What is the least number which when divided by 8,12,15 and 20 leaves in each…
2025
What is the least number which when divided by 8,12,15 and 20 leaves in each case a remainder of 5 ?
- A.
125
- B.
117
- C.
132
- D.
112
Attempted by 31 students.
Show answer & explanation
Correct answer: A
Key idea: the required number N must satisfy N ≡ 5 (mod 8), N ≡ 5 (mod 12), N ≡ 5 (mod 15), and N ≡ 5 (mod 20). Therefore N − 5 must be divisible by 8, 12, 15 and 20.
Compute the LCM of 8, 12, 15 and 20.
Prime factorizations: 8 = 2³, 12 = 2² × 3, 15 = 3 × 5, 20 = 2² × 5. Take the highest powers: 2³ × 3 × 5 = 120.
Since N − 5 must be a multiple of 120, write N = 120k + 5. The smallest positive integer k is 1, so N = 120 × 1 + 5 = 125.
Check: 125 mod 8 = 5, 125 mod 12 = 5, 125 mod 15 = 5, 125 mod 20 = 5. All conditions are satisfied.
Answer: 125