A number when divided by 45 leaves remainder 27, and when divided by 27 leaves remainder 9. Find the smallest such number.
Show answer & explanation
Correct answer: B
Explanation
Given:
N divided by 45 leaves a remainder of 27
N divided by 27 leaves a remainder of 9
By the formula:
If N divided by a leaves a remainder of r, then
N = aq + r
N = 45a + 27
45a + 27 ≡ 9 (mod 27)
45a + 18 ≡ 0 (mod 27)
45a ≡ −18 ≡ 9 (mod 27)
45 ≡ 18 (mod 27)
18a ≡ 9 (mod 27)
Dividing both sides by 9
2a ≡ 1 (mod 3)
Substituting a = 2 satisfies
N = 45(2) + 27
= 90 + 27
= 117