The smallest no. such that if its rightmost digit is placed at its left end, the new number so formed is precisely 50% larger than the original no.

Ques Find the smallest number such that if its rightmost digit is placed at its left end, the new number so formed is precisely 50% larger than the original number.
Answer The answer is 285714. If its rightmost digit is placed at its left end, then new number is 428571 which is 50% larger than the original number 285714.

Sum of last digit = sum of rest digits

Consider a number 235, where last digit is the sum of first two digits i.e. 2 + 3 = 5. How many such 3-digit numbers are there?
Ans There are 45 different 3-digit numbers.

Take the combinations
The last digit can not be 0.
If the last digit is 1, the only possible number is 101. (Note that 011 is not a 3-digit number)
If the last digit is 2, the possible numbers are 202 and 112.
If the last digit is 3, the possible numbers are 303, 213 and 123.
If the last digit is 4, the possible numbers are 404, 314, 224 and 134.
If the last digit is 5, the possible numbers are 505, 415, 325, 235 and 145.

Note the pattern here - If the last digit is 1, there is only one number. If the last digit is 2, there are two numbers. If the last digit is 3, there are three numbers. If the last digit is 4, there are four numbers. If the last digit is 5, there are five numbers. And so on.....
Thus, total numbers are 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45

Altogether then, there are 45 different 3-digit numbers, where last digit is the sum of first two digits.

Special numbers

A number of 9 digits has the following properties:

• The number comprising the leftmost two digits is divisible by 2, that comprising the leftmost three digits is divisible by 3, the leftmost four by 4, the leftmost five by 5, and so on for the nine digits of the number i.e. the number formed from the first n digits is divisible by n, 2<=n<=9.

• Each digit in the number is different i.e. no digits are repeated.

• The digit 0 does not occur in the number i.e. it is comprised only of the digits 1-9 in some order.

Find the number.

Answer
The answer is 381654729 One way to solve it is Trial-&-Error. You can make it bit easier as odd positions will always occupy ODD numbers and even positions will always occupy EVEN numbers. Further 5th position will contain 5 as 0 does not occur.
At odd points we have  3 1 5 7 9
At even points we have 8 6 4 2

Armstrong number
Armstrong numbers are the sum of their own digits to the power of the number of digits. example:153 = 1³ + 5³ + 3³[as sum of the digits are 3], 1⁴ + 6⁴ + 3⁴ + 4⁴ = 1634.