Question

A certain number, when divided by 899, leaves a remainder 63. Find the remainder when the same number is divided by 29.

• A. 5
• B. 4
• C. 3
• D. Cannot be determined

Hint:

When finding remainders after division by a smaller number, reduce intermediate values modulo the smaller divisor to simplify the calculations.

Answer 1

The Correct Option is A

Let the number be \( N \). We are given that: \[ N = 899k + 63, \] for some integer \( k \). Now, we want to find the remainder when \( N \) is divided by 29. First, reduce 899 modulo 29: \[ 899 \div 29 = 31 \text{ (quotient)} \quad \text{and} \quad 899 - 29 \times 31 = 899 - 899 = 0. \] Thus, \( 899 \equiv 0 \, (\text{mod} \, 29) \). Hence: \[ N = 899k + 63 \equiv 0k + 63 \equiv 63 \, (\text{mod} \, 29). \] Now, reduce 63 modulo 29: \[ 63 \div 29 = 2 \quad \text{(quotient)} \quad \text{and} \quad 63 - 29 \times 2 = 63 - 58 = 5. \] Thus, the remainder when \( N \) is divided by 29 is 5.