Remainders, the Minus-One Trick, Casting Out Nines and Binary-Decimal Conversion

Remainders: Big Numbers, Tiny Answers

Questions such as 'what is the remainder when 7 × 13 × 19 is divided by 5?' or 'when 2^100 is divided by 3?' look frightening, yet they never require the full product. Every division obeys one identity: Dividend = Divisor × Quotient + Remainder, with the remainder always smaller than the divisor. The big idea is that remainders can be taken early. You may replace every number by its own remainder before you add or multiply, and the final remainder comes out the same. This short gives you the product rule, the minus-one trick for powers, the digit-sum check for 9 and 3, and the binary-to-decimal conversion that SSC and Railway papers love. Each method turns a page of arithmetic into two lines.

1. What is the remainder when 17 × 23 is divided by 5?

2. What is the remainder when 2^100 is divided by 3?

3. What is the remainder when 5^99 is divided by 6?

4. The binary number 1101 is equal to which decimal number?

5. What is the binary form of the decimal number 25?

6. What is the remainder when 123456 is divided by 9?

7. What is the least number which leaves remainder 5 when divided by 8 and also by 12?

8. A number leaves remainder 4 when divided by 7. What remainder does its square leave when divided by 7?