Remainder / Division / Modular Arithmetic

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Q. When a number is divided by 899, the remainder is 63. What will be the remainder when the same number is divided by 29 ?

Options: (A) 3 (B) 7 (C) 1 (D) 5

Solⁿ:  899 = 29 × 31 So, dividing by 899 means dividing by a multiple of 29.

The number can be thought of as: (something divisible by 29) + 63

Now, just divide 63 by 29: 29 × 2 = 58, 63 − 58 = 5

Ans: 5

Trick: If a number leaves remainder r when divided by a multiple of c, then remainder when divided by c = r ÷ c remainder. So here: Remainder when divided by 29 = 63 ÷ 29 = 5

Q. A number leaves remainder 45 when divided by 127. What will be the remainder when the same number is divided by 7 ?

Solⁿ:  If 127 is a multiple of 7 : 127 ÷ 7 = 18 remainder 1 → still fine, we only focus on the remainder.

Number can be written as 127x + 45

Divide 45 by 7: 7 × 6 = 42, 45 − 42 = 3

Ans: 3

Q. When a number is divided by 221, the remainder is 87. What is the remainder when the same number is divided by 13 ?

Solⁿ : 221 ÷ 13 = 17 , so 221 = 13 × 17, Number = 221x + 87 → 13 × 17x part divisible by 13 → remainder = 0

Divide 87 by 13: 13 × 6 = 78, 87 − 78 = 9

Ans: 9

Q. A number leaves remainder 125 when divided by 396. What will be the remainder when the same number is divided by 12 ?

Solⁿ: 396 ÷ 12 = 33 → exact, so 396 = 12 × 33, Number = 396k + 125 → 12 × 33k divisible by 12 → remainder = 0

Divide 125 by 12: 12 × 10 = 120, 125 − 120 = 5

Ans: 5

Key Trick : If a number leaves remainder r when divided by a multiple of some number c, then the remainder when divided by c = r ÷ c remainder. Only the remainder part matters when dividing by a factor of the original divisor. This trick saves a lot of time!