Factorial upto 10

n	n!
0	1
1	1
2	2
3	6
4	24
5	120
6	720
7	5,040
8	40,320
9	362,880
10	3,628,800

Conversion : Distance and weight

1 mile = 1760 yards
1 yard = 3 feet
1 mile² = 640 acres
I gallon = 4 quarts
1 quart = 2 pints
1 pint = 2 cups
1 cup = 8 ounces
1 pound = 16 ounces
1 ounce = 16 drams
1 kg = 2.2 pounds

Conversion : Distance and weight

Distance
1 mile = 1760 yards
1 yard = 3 feet
1 mile = 1.6 km (nearly)

Area
1 mile² = 640 acres

Speed
1 km/hr = 5/18 m/sec
1 m/sec = 18/5 km/hr

Volume
I gallon = 4 quarts
1 quart = 2 pints
1 pint = 2 cups
1 cup = 8 ounces
1 pound = 16 ounces
1 ounce = 16 drams
1 kg = 2.2 pounds

Conversion : Money

Dollar
1 Nickel = 5 cents
1 dime = 10 cents
1 quarter = 25 cents
1 half = 50 cents
1 dollar = 100 cents

CALENDAR notes

• Calendar repeats after every 400 years.
• Leap year- it is always divisible by 4, but century years are not leap years unless they are divisible by 400.
• Century has 5 odd days and leap century has 6 odd days.
• In a normal year 1st January and 2nd July and 1st October fall on the same day. In a leap year 1st January 1st July and 30th September fall on the same day.
• January 1, 1901 was a Tuesday.
  

Cram misc

 

Certain numbers which didn't occurred in previous cramming posts are here:

 

• 210 = 45 = 322 = 1024
• 38 = 94 = 812 = 6561
• 7 * 11 * 13 = 1001
• 11 * 13 * 17 = 2431
• 13 * 17 * 19 = 4199
• 19 * 21 * 23 = 9177
• 19 * 23 * 29 = 12673
  

Square, cube, reciprocals and roots of first 30 numbers

Here are values to be crammed for making fast calculations :

	Reciprocal	square	cube	roots
1	1.00	1	1	1
2	0.50	4	8	1.41421
3	0.33	9	27	1.73205
4	0.25	16	64	2
5	0.20	25	125	2.23607
6	0.16	36	216	2.44949
7	0.142857	49	343	2.64575
8	0.125	64	512	2.82843
9	0.11	81	729	3
10	0.10	100	1000	3.16228
11	0.09	121	1331	3.31662
12	0.083	144	1728	3.4641
13	0.0769	169	2197	3.60555
14	0.0714286	196	2744	3.74166
15	0.066	225	3375	3.87298
16	0.0625	256	4096	4
17	0.0588	289	4913	4.12311
18	0.055	324	5832	4.24264
19	0.0526	361	6859	4.3589
20	0.05	400	8000	4.47214
21	0.0476	441	9261	4.58258
22	0.045	484	10648	4.69042
23	0.0434	529	12167	4.79583
24	0.0416	576	13824	4.89898
25	0.04	625	15625	5
26	0.0385	676	17576	5.09902
27	0.037	729	19683	5.19615
28	0.0357	784	21952	5.2915
29	0.0344	841	24389	5.38516
30	0.033	900	27000	5.47723

Some Pythagoras triplets to cram

In any given exam there are about 2 to 3 questions based on Pythagoras theorem.  Wouldn't it be nice that you remember some of the Pythagoras triplets thus saving up to 30 seconds in each question. This saved time may be used to attempt other questions. Remember one more right question may make a lot of difference in UR PERCENTILE score.The unique set of Pythagoras triplets with the Hypotenuse less than 100 or one of the side less than 20  are as follows :

(3,4,5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29), (12, 35, 37), (9, 40, 41), (28, 45, 53), (11, 60, 61), (33, 56, 65), (16, 63, 65), (48, 55, 73), (36, 77, 85), (13, 84, 85), (39, 80, 89), and (65, 72, 97)….even more…
(15,112,113),   (17,144,145),   (19,180,181),   (20,99,101)

If you multiply the digits of the above mentioned sets by any constant you will again get a Pythagoras triplet .
Example : Take the set (3,4,5).
Multiply it by 2 you get (6,8,10) which is also a pythagoras triplet.
Multiply it by 3 you get ( 9,12,15) which is also a pythagoras triplet.
Multiply it by 4 you get (12,16,20) which is also a pythagoras triplet.
You may multiply by any constant you will get a pythagoras triplet
Take another example (5,12,13)
Multiply it by 5,6 and 7 and check if you get a pythagoras triplet.

TIPS FOR SMART GUESSING :
You will notice that in any case, whether it is a unique triplet or it is a derived triplet (derived by multiplying a constant to a unique triplet), all the three numbers cannot be odd. 
In case of unique triplet , the hypotenuse is always odd and one of the remaining side is odd the other one is even.
Below are the first few unique triplets with first number as Odd.
3    4    5
5    12   13
7    24   25
9    40   41
11   60   61
You will notice following trend for unique triplets with first side as odd.
Hypotenuse = (Sq(first side) +1) / 2
Other side = Hypotenuse –1 or (first side * n + n)

Example : First side = 3 ,
so hypotenuse = (3*3+1)/2= 5 and other side = 5-1=4
Example 2: First side = 11
so hypotenuse = (9*9+1)/2= 41 and other side = 41-1=40

Please note that the above is not true for a derived  triplet for example 9,12 and 15, which has been obtained from multiplying 3 to the triplet of 3,4,5.  You may check for other derived triplets.
Below are the first few unique triplets with first number as Even .
4    3    5
8    15   17
12   35   37
16   63   65
20   99   101
You will notice following trend for unique triplets with first side as Even.
Hypotenuse = Sq( first side/ 2)+1
Other side = Hypotenuse-2

Example 1. First side =8
So hypotenuse = sq(8/2) +1= 17
Other side = 17-2=15
Example 2. First side = 16
So hypotenuse = Sq(16/2) +1 =65
Other side = 65-2= 63

Cubes upto 30

Following are the cubes of first 30 numbers :

1	1
2	8
3	27
4	64
5	125
6	216
7	343
8	512
9	729
10	1000
11	1331
12	1728
13	2197
14	2744
15	3375
16	4096
17	4913
18	5832
19	6859
20	8000
21	9261
22
23
24
25
26
27
28
29
30	27000

Vedic mathematics : Easy way of finding square of a number ending with 5

quick way to square numbers that end in 5 using the formula BY ONE MORE THAN THE ONE BEFORE.

• 75² = 5625 75² means 75 x 75.
  The answer is in two parts: 56 and 25.
  The last part is always 25.
  The first part is the first number, 7, multiplied by the number "one more", which is 8:
  so 7 x 8 = 56
• Similarly 85² = (8 * 9) 25 = 7225
  

Square of 2 digit number having same digit, AA

N     N^2
11   121
22   484
33   1089
44   1936
55   2916
66   4356
77   5776
88   7744
99   9801


Now suppose that number is AA
than AA = 10A+A
We know,
(a+b)^2 = a^2 + 2ab + b^2
AA ^ 2 = (10A +A) ^2 = 100*(A^2 ) + (A^2 ) + 2*10A*A= 121*(A^2 )
So all these numbers are divided by 121 :P

So if you want to find 99^2, you can do 121 * (9^2), though it may look tough this way.
=121 * 81
But for 22 ^2 = 121 * 4 = 484 , i.e. little easier

Squares upto 100

Number	Square
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144
13	169
14	196
15	225
16	256
17	289
18	324
19	361
20	400
21	441
22	484
23	529
24	576
25	625
26	676
27	729
28	784
29	841
30	900
31	961
32	1024
33	1089
34	1156
35	1225
36	1296
37	1369
38	1444
39	1521
40	1600
41	1681
42	1764
43	1849
44	1936
45	2025
46	2116
47	2209
48	2304
49	2401
50	2500
51	2601
52	2704
53	2809
54	2916
55	3025
56	3136
57	3249
58	3364
59	3481
60	3600
61	3721
62	3844
63	3969
64	4096
65	4225
66	4356
67	4489
68	4624
69	4761
70	4900
71	5041
72	5184
73	5329
74	5476
75	5625
76	5776
77	5929
78	6084
79	6241
80	6400
81	6561
82	6724
83	6889
84	7056
85	7225
86	7396
87	7569
88	7744
89	7921
90	8100
91	8281
92	8464
93	8649
94	8836
95	9025
96	9216
97	9409
98	9604
99	9801
100	10000

Quant… Basic Formulae

Consolidated some of the basic formula.
ALGEBRA :
1. Sum of first n natural numbers = n(n+1)/2
2. Sum of the squares of first n natural numbers = n(n+1)(2n+1)/6
3. Sum of the cubes of first n natural numbers = [n(n+1)/2]2
4. Sum of first n natural odd numbers = n2
5. Average = (Sum of items)/Number of items
Arithmetic Progression (A.P.):
An A.P. is of the form a, a+d, a+2d, a+3d, …
where a is called the ‘first term’ and d is called the ‘common difference’
1. nth term of an A.P. tn = a + (n-1)d
2. Sum of the first n terms of an A.P. Sn = n/2[2a+(n-1)d] or Sn = n/2(first term + last term)
Geometrical Progression (G.P.):
A G.P. is of the form a, ar, ar2, ar3, …
where a is called the ‘first term’ and r is called the ‘common ratio’.
1. nth term of a G.P. tn = arn-1
2. Sum of the first n terms in a G.P. Sn = a|1-rn|/|1-r|
Permutations and Combinations :
1. nPr = n!/(n-r)!
2. nPn = n!
3. nP1 = n
1. nCr = n!/(r! (n-r)!)
2. nC1 = n
3. nC0 = 1 = nCn
4. nCr = nCn-r
5. nCr = nPr/r!
Number of diagonals in a geometric figure of n sides = nC2-n

Tests of Divisibility :
1. A number is divisible by 2 if it is an even number.
2. A number is divisible by 3 if the sum of the digits is divisible by 3.
3. A number is divisible by 4 if the number formed by the last two digits is divisible by 4.
4. A number is divisible by 5 if the units digit is either 5 or 0.
5. A number is divisible by 6 if the number is divisible by both 2 and 3.
6. A number is divisible by 8 if the number formed by the last three digits is divisible by 8.
7. A number is divisible by 9 if the sum of the digits is divisible by 9.
8. A number is divisible by 10 if the units digit is 0.
9. A number is divisible by 11 if the difference of the sum of its digits at odd places and the sum of its digits at even places, is divisible by 11.
H.C.F and L.C.M :
H.C.F stands for Highest Common Factor. The other names for H.C.F are Greatest Common Divisor (G.C.D) and Greatest Common Measure (G.C.M).
The H.C.F. of two or more numbers is the greatest number that divides each one of them exactly.
The least number which is exactly divisible by each one of the given numbers is called their L.C.M.
Two numbers are said to be co-prime if their H.C.F. is 1.
H.C.F. of fractions = H.C.F. of numerators/L.C.M of denominators
L.C.M. of fractions = G.C.D. of numerators/H.C.F of denominators
Product of two numbers = Product of their H.C.F. and L.C.M.

PERCENTAGES :
1. If A is R% more than B, then B is less than A by R / (100+R) * 100
2. If A is R% less than B, then B is more than A by R / (100-R) * 100
3. If the price of a commodity increases by R%, then reduction in consumption, not to increase the expenditure is : R/(100+R)*100
4. If the price of a commodity decreases by R%, then the increase in consumption, not to decrease the expenditure is : R/(100-R)*100
PROFIT & LOSS :
1. Gain = Selling Price(S.P.) – Cost Price(C.P)
2. Loss = C.P. – S.P.
3. Gain % = Gain * 100 / C.P.
4. Loss % = Loss * 100 / C.P.
5. S.P. = (100+Gain%)/100*C.P.
6. S.P. = (100-Loss%)/100*C.P.
Short cut Methods:
1. By selling an article for Rs. X, a man loses l%. At what price should he sell it to gain y%? (or)
A man lost l% by selling an article for Rs. X. What percent shall he gain or lose by selling it for Rs. Y?
(100 – loss%) : 1st S.P. = (100 + gain%) : 2nd S.P.
2. A man sold two articles for Rs. X each. On one he gains y% while on the other he loses y%. How much does he gain or lose in the whole transaction?
In such a question, there is always a lose. The selling price is immaterial.
Formula: Loss % =
3. A discount dealer professes to sell his goods at cost price but uses a weight of 960 gms. For a kg weight. Find his gain percent.
Formula: Gain % =
RATIO & PROPORTIONS:
1. The ratio a : b represents a fraction a/b. a is called antecedent and b is called consequent.
2. The equality of two different ratios is called proportion.
3. If a : b = c : d then a, b, c, d are in proportion. This is represented by a : b :: c : d.
4. In a : b = c : d, then we have a* d = b * c.
5. If a/b = c/d then ( a + b ) / ( a – b ) = ( d + c ) / ( d – c ).
TIME & WORK :
1. If A can do a piece of work in n days, then A’s 1 day’s work = 1/n
2. If A and B work together for n days, then (A+B)’s 1 days’s work = 1/n
3. If A is twice as good workman as B, then ratio of work done by A and B = 2:1
PIPES & CISTERNS :
1. If a pipe can fill a tank in x hours, then part of tank filled in one hour = 1/x
2. If a pipe can empty a full tank in y hours, then part emptied in one hour = 1/y
3. If a pipe can fill a tank in x hours, and another pipe can empty the full tank in y hours, then on opening both the pipes,
the net part filled in 1 hour = (1/x-1/y) if y>x
the net part emptied in 1 hour = (1/y-1/x) if x>y
TIME & DISTANCE :
1. Distance = Speed * Time
2. 1 km/hr = 5/18 m/sec
3. 1 m/sec = 18/5 km/hr
4. Suppose a man covers a certain distance at x kmph and an equal distance at y kmph. Then, the average speed during the whole journey is 2xy/(x+y) kmph.
PROBLEMS ON TRAINS :
1. Time taken by a train x metres long in passing a signal post or a pole or a standing man is equal to the time taken by the train to cover x metres.
2. Time taken by a train x metres long in passing a stationary object of length y metres is equal to the time taken by the train to cover x+y metres.
3. Suppose two trains are moving in the same direction at u kmph and v kmph such that u>v, then their relative speed = u-v kmph.
4. If two trains of length x km and y km are moving in the same direction at u kmph and v kmph, where u>v, then time taken by the faster train to cross the slower train = (x+y)/(u-v) hours.
5. Suppose two trains are moving in opposite directions at u kmph and v kmph. Then, their relative speed = (u+v) kmph.
6. If two trains of length x km and y km are moving in the opposite directions at u kmph and v kmph, then time taken by the trains to cross each other = (x+y)/(u+v)hours.
7. If two trains start at the same time from two points A and B towards each other and after crossing they take a and b hours in reaching B and A respectively, then A’s speed : B’s speed = (√b : √
SIMPLE & COMPOUND INTERESTS :
Let P be the principal, R be the interest rate percent per annum, and N be the time period.
1. Simple Interest = (P*N*R)/100
2. Compound Interest = P(1 + R/100)N – P
3. Amount = Principal + Interest
LOGORITHMS :
If am = x , then m = logax.
Properties :
1. log xx = 1
2. log x1 = 0
3. log a(xy) = log ax + log ay
4. log a(x/y) = log ax – log ay
5. log ax = 1/log xa
6. log a(xp) = p(log ax)
7. log ax = log bx/log ba
Note : Logarithms for base 1 does not exist.
AREA & PERIMETER :
Shape Area Perimeter
Circle ∏ (Radius)2 2∏(Radius)
Square (side)2 4(side)
Rectangle length*breadth 2(length+breadth)
1. Area of a triangle = 1/2*Base*Height or
2. Area of a triangle = √ (s(s-(s-b)(s-c)) where a,b,c are the lengths of the sides and s = (a+b+c)/2
3. Area of a parallelogram = Base * Height
4. Area of a rhombus = 1/2(Product of diagonals)
5. Area of a trapezium = 1/2(Sum of parallel sides)(distance between the parallel sides)
6. Area of a quadrilateral = 1/2(diagonal)(Sum of sides)
7. Area of a regular hexagon = 6(√3/4)(side)2
8. Area of a ring = ∏(R2-r2) where R and r are the outer and inner radii of the ring.
VOLUME & SURFACE AREA :
Cube :
Let a be the length of each edge. Then,
1. Volume of the cube = a3 cubic units
2. Surface Area = 6a2 square units
3. Diagonal = √ 3 a units
Cuboid :
Let l be the length, b be the breadth and h be the height of a cuboid. Then
1. Volume = lbh cu units
2. Surface Area = 2(lb+bh+lh) sq units
3. Diagonal = √ (l2+b2+h2)
Cylinder :
Let radius of the base be r and height of the cylinder be h. Then,
1. Volume = ∏r2h cu units
2. Curved Surface Area = 2∏rh sq units
3. Total Surface Area = 2∏rh + 2∏r2 sq units
Cone :
Let r be the radius of base, h be the height, and l be the slant height of the cone. Then,
1. l2 = h2 + r2
2. Volume = 1/3(∏r2h) cu units
3. Curved Surface Area = ∏rl sq units
4. Total Surface Area = ∏rl + ∏r2 sq units
Sphere :
Let r be the radius of the sphere. Then,
1. Volume = (4/3)∏r3 cu units
2. Surface Area = 4∏r2 sq units
Hemi-sphere :
Let r be the radius of the hemi-sphere. Then,
1. Volume = (2/3)∏r3 cu units
2. Curved Surface Area = 2∏r2 sq units
3. Total Surface Area = 3∏r2 sq units
Prism :
Volume = (Area of base)(Height

Some useful fractions to learn

we have to calculate 1/x when x is
x    1/x 
2     .5
3     .{3}
4     .25
5     .20
6     .1{6}
7     .{142857}
8     .125
9     .{1}
10   .1
11    .{09}

Divisibility tests

A number is divisible by 2 if its last digit is also (i.e. 0,2,4,6 or 8).

A number is divisible by 3 if the sum of its digits is also. Example: 534: 5+3+4=12 and 1+2=3 so 534 is divisible by 3.

A number is divisible by 4, if last 2 numbers are divisible by 4. Likewise the number is divisible by 8, if its last 3 digits are divisible by 8. So this holds for powers of 2.

A number is divisible by 5 if the last digit is 5 or 0.

Most people know (only) those 3 rules. Here are the rules for divisibility by the PRIMES up to 50. Why only primes and not also composite numbers? A number is divisible by a composite if it is also divisible by all the prime factors (e.g. is divisible by 21 if divisible by 3 AND by 7). Small numbers are used in these worked examples, so you could have used a pocket calculator. But my rules apply to any number of digits, whereas you cannot test a 30 or more digit number on your pocket calculator otherwise.
Lets assume L is the last digit and A is the remaining truncating number....

Test for divisibility by 7. Double the last digit and subtract it from the remaining leading truncated number. If the result is divisible by 7, then so was the original number. Apply this rule over and over again as necessary. Example: 826. Twice 6 is 12. So take 12 from the truncated 82. Now 82-12=70. This is divisible by 7, so 826 is divisible by 7 also.
There are similar rules for the remaining primes under 40, i.e. 11,13, 17,19,23,29,31,37,41,43 and 47.
(A-2L) / 7


Test for divisibility by 11. Subtract the last digit from the remaining leading truncated number. If the result is divisible by 11, then so was the first number. Apply this rule over and over again as necessary.
Example: 19151--> 1915-1 =1914 -->191-4=187 -->18-7=11, so yes, 19151 is divisible by 11.
(A-L) / 11
     Another approach is to sum up numbers in 2 parts - 1st part containing all odd positioned numbers and other part containing all numbers positioned at  even positions. Find the difference and if the difference is divided by 11, i.e. if it is 11 or 0 it means number is divided by 11. eg. 19151 ,
odd position sum = 3, even position sum = 14; Difference = 11, that means no. is divisible by 11.

Test for divisibility by 13. Add four times the last digit to the remaining leading truncated number. If the result is divisible by 13, then so was the first number. Apply this rule over and over again as necessary.
Example: 50661-->5066+4=5070-->507+0=507-->50+28=78 and 78 is 6*13, so 50661 is divisible by 13.

(A+4L) / 13

Test for divisibility by 17. Subtract five times the last digit from the remaining leading truncated number. If the result is divisible by 17, then so was the first number. Apply this rule over and over again as necessary.
Example: 3978-->397-5*8=357-->35-5*7=0. So 3978 is divisible by 17.
(A-5L) / 17




  Test for divisibility by 19. Add two times the last digit to the remaining leading truncated number. If the result is divisible by 19, then so was the first number. Apply this rule over and over again as necessary.
EG: 101156-->10115+2*6=10127-->1012+2*7=1026-->102+2*6=114 and 114=6*19, so 101156 is divisible by 19.
(A+2L) / 19

I think what I have above written is sufficient..but if you want more rules please carry on... :)

Test for divisibility by 23. Add seven times the last digit to the remaining leading truncated number. If the result is divisible by 23, then so was the first number. Apply this rule over and over again as necessary.
Example: 17043-->1704+7*3=1725-->172+7*5=207-->20+7*7=69 which is 3*23, so 17043 is also divisible by 23.
(A+7L) / 23




Test for divisibility by 29. Add three times the last digit to the remaining leading truncated number. If the result is divisible by 29, then so was the first number. Apply this rule over and over again as necessary.
Example: 15689-->1568+3*9=1595-->159+3*5=174-->17+3*4=29, so 15689 is also divisible by 29.
(A+3L) / 29


Test for divisibility by 31. Subtract three times the last digit from the remaining leading truncated number. If the result is divisible by 31, then so was the first number. Apply this rule over and over again as necessary.
Example: 7998-->799-3*8=775-->77-3*5=62 which is twice 31, so 7998 is also divisible by 31.
(A-3L) / 31

Test for divisibility by 37. This is (slightly) more difficult, since it perforce uses a double-digit multiplier, namely eleven. People can usually do single digit multiples of 11, so we can use the same technique still. Subtract eleven times the last digit from the remaining leading truncated number. If the result is divisible by 37, then so was the first number. Apply this rule over and over again as necessary.
Example: 23384-->2338-11*4=2294-->229-11*4=185 which is five times 37, so 23384 is also divisible by 37.
(A-11L) / 37


Test for divisibility by 41. Subtract four times the last digit from the remaining leading truncated number. If the result is divisible by 41, then so was the first number. Apply this rule over and over again as necessary.
Example: 30873-->3087-4*3=3075-->307-4*5=287-->28-4*7=0, remainder is zero and so 30873 is also divisible by 41.
(A+2L) / 41


Test for divisibility by 43. Now it starts to get really difficult for most people, because the multiplier to be used is 13, and most people cannot recognise even single digit multiples of 13 at sight. You may want to make a little list of 13*N first. Nevertheless, for the sake of completeness, we will use the same method. Add thirteen times the last digit to the remaining leading truncated number. If the result is divisible by 43, then so was the first number. Apply this rule over and over again as necessary.
Example: 3182-->318+13*2=344-->34+13*4=86 which is recognisably twice 43, and so 3182 is also divisible by 43.
(A+13L) / 43


Test for divisibility by 47. This too is difficult for most people, because the multiplier to be used is 14, and most people cannot recognise even single digit multiples of 14 at sight. You may want to make a little list of 14*N first. Nevertheless, for the sake of completeness, we will use the same method. Subtract fourteen times the last digit from the remaining leading truncated number. If the result is divisible by 47, then so was the first number. Apply this rule over and over again as necessary.
Example: 34827-->3482-14*7=3384-->338-14*4=282-->28-14*2=0 , remainder is zero and so 34827 is divisible by 47.
(A+2L) / 19


I've stopped here at the last prime below 50, for arbitrary but pragmatic reasons as explained above.
Lets summarize A+mL divisibilities :

Number	Coeff of L (m)
7	-2
11	-1
13	4
17	-5
19	2
23	7
29	3
31	-3
37	-11
41	2
43	13
47	-14

Other blogreaders (sadly even people from .edu domains, who should be able to do the elementary algebra themselves) have asked why I sometimes say ADD and for other primes say SUBTRACT, and ask where the apparently arbitrary factors come from. So let us do some algebra to show the method in my madness.
We have displayed the recursive divisibility test of number N as f-M*r where f are the front digits of N, r is the rear digit of N and M is some multiplier. And we want to see if N is divisible by some prime P. We need a method to work out the values of M. What you do is to calculate (mentally) the smallest multiple of P which ends in a 9 or a 1. If it's a 9 we are going to ADD, if it's a 1 we are going to SUBTRACT later. Then we will use the leading digit(s) of the multiple as our multiplier M.
Example for P=17 : three times 17 is 51 which is the smallest multiple of 17 that ends in a 1 or 9. Since it's a 1 we are going to SUBTRACT later. The leading digit is a 5, so we are going to SUBTRACT five times the remainder r. The algorithm was stated above. Now let's do the algebraic proof. Writing N=10f+r, we can multiply by -5 (as shown in the example for 17), getting -5N=-50f-5r. Now we add 51f to both sides (because 51 was the smallest multiple of P=17 to end in a 1 or a 9), giving one f (which we want), so 51f-5N=f-5r. Now if N is divisible by P (here P=17), we can substitute to get 51f-5*17*x=f-5r and rearrange the left side as 17*(3f-5x)=f-5r and therefore f-5r is a multiple of P=17 also. Q.E.D.

Powers

Multiplication Tables

1 × 1 = 1	2 × 1 = 2 2 × 2 = 4	3 × 1 = 3 3 × 2 = 6 3 × 3 = 9	4 × 1 = 4 4 × 2 = 8 4 × 3 = 12 4 × 4 = 16	5 × 1 = 5 5 × 2 = 10 5 × 3 = 15 5 × 4 = 20 5 × 5 = 25
6 × 1 = 6 6 × 2 = 12 6 × 3 = 18 6 × 4 = 24 6 × 5 = 30 6 × 6 = 36	7 × 1 = 7 7 × 2 = 14 7 × 3 = 21 7 × 4 = 28 7 × 5 = 35 7 × 6 = 42 7 × 7 = 49	8 × 1 = 8 8 × 2 = 16 8 × 3 = 24 8 × 4 = 32 8 × 5 = 40 8 × 6 = 48 8 × 7 = 56 8 × 8 = 64	9 × 1 = 9 9 × 2 = 18 9 × 3 = 27 9 × 4 = 36 9 × 5 = 45 9 × 6 = 54 9 × 7 = 63 9 × 8 = 72 9 × 9 = 81	10 × 1 = 10 10 × 2 = 20 10 × 3 = 30 10 × 4 = 40 10 × 5 = 50 10 × 6 = 60 10 × 7 = 70 10 × 8 = 80 10 × 9 = 90 10 × 10 = 100
11 × 1 = 11 11 × 2 = 22 11 × 3 = 33 11 × 4 = 44 11 × 5 = 55 11 × 6 = 66 11 × 7 = 77 11 × 8 = 88 11 × 9 = 99 11 × 10 = 110 11 × 11 = 121	12 × 1 = 12 12 × 2 = 24 12 × 3 = 36 12 × 4 = 48 12 × 5 = 60 12 × 6 = 72 12 × 7 = 84 12 × 8 = 96 12 × 9 = 108 12 × 10 = 120 12 × 11 = 132 12 × 12 = 144	13 × 1 = 13 13 × 2 = 26 13 × 3 = 39 13 × 4 = 52 13 × 5 = 65 13 × 6 = 78 13 × 7 = 91 13 × 8 = 104 13 × 9 = 117 13 × 10 = 130 13 × 11 = 143 13 × 12 = 156 13 × 13 = 169	14 × 1 = 14 14 × 2 = 28 14 × 3 = 42 14 × 4 = 56 14 × 5 = 70 14 × 6 = 84 14 × 7 = 98 14 × 8 = 112 14 × 9 = 126 14 × 10 = 140 14 × 11 = 154 14 × 12 = 168 14 × 13 = 182 14 × 14 = 196	15 × 1 = 15 15 × 2 = 30 15 × 3 = 45 15 × 4 = 60 15 × 5 = 75 15 × 6 = 90 15 × 7 = 105 15 × 8 = 120 15 × 9 = 135 15 × 10 = 150 15 × 11 = 165 15 × 12 = 180 15 × 13 = 195 15 × 14 = 210 15 × 15 = 225
16 × 1 = 16 16 × 2 = 32 16 × 3 = 48 16 × 4 = 64 16 × 5 = 80 16 × 6 = 96 16 × 7 = 112 16 × 8 = 128 16 × 9 = 144 16 × 10 = 160 16 × 11 = 176 16 × 12 = 192 16 × 13 = 208 16 × 14 = 224 16 × 15 = 240 16 × 16 = 256	17 × 1 = 17 17 × 2 = 34 17 × 3 = 51 17 × 4 = 68 17 × 5 = 85 17 × 6 = 102 17 × 7 = 119 17 × 8 = 136 17 × 9 = 153 17 × 10 = 170 17 × 11 = 187 17 × 12 = 204 17 × 13 = 221 17 × 14 = 238 17 × 15 = 255 17 × 16 = 272 17 × 17 = 289	18 × 1 = 18 18 × 2 = 36 18 × 3 = 54 18 × 4 = 72 18 × 5 = 90 18 × 6 = 108 18 × 7 = 126 18 × 8 = 144 18 × 9 = 162 18 × 10 = 180 18 × 11 = 198 18 × 12 = 216 18 × 13 = 234 18 × 14 = 252 18 × 15 = 270 18 × 16 = 288 18 × 17 = 306 18 × 18 = 324	19 × 1 = 19 19 × 2 = 38 19 × 3 = 57 19 × 4 = 76 19 × 5 = 95 19 × 6 = 114 19 × 7 = 133 19 × 8 = 152 19 × 9 = 171 19 × 10 = 190 19 × 11 = 209 19 × 12 = 228 19 × 13 = 247 19 × 14 = 266 19 × 15 = 285 19 × 16 = 304 19 × 17 = 323 19 × 18 = 342 19 × 19 = 361	20 × 1 = 20 20 × 2 = 40 20 × 3 = 60 20 × 4 = 80 20 × 5 = 100 20 × 6 = 120 20 × 7 = 140 20 × 8 = 160 20 × 9 = 180 20 × 10 = 200 20 × 11 = 220 20 × 12 = 240 20 × 13 = 260 20 × 14 = 280 20 × 15 = 300 20 × 16 = 320 20 × 17 = 340 20 × 18 = 360 20 × 19 = 380 20 × 20 = 400