How many integers from 1 to 250 (inclusive) are divisible by 2, 5, and 7?
2010
How many integers from 1 to 250 (inclusive) are divisible by 2, 5, and 7?
Answer: B. 3 — Concept: An integer divisible by several numbers must be a multiple of their least common multiple (LCM). The number of positive multiples of d not exceeding…
- A.
2
- B.
3
- C.
5
- D.
8
Attempted by 5 students.
Show answer & explanation
Correct answer: B
Concept: An integer divisible by several numbers must be a multiple of their least common multiple (LCM). The number of positive multiples of d not exceeding N is ⌊N/d⌋.
Application
Because 2, 5, and 7 are distinct prime numbers, LCM(2, 5, 7) = 2 × 5 × 7 = 70.
The positive multiples of 70 in the interval are 70, 140, and 210; the next multiple, 280, exceeds 250.
Equivalently, ⌊250/70⌋ = 3.
Cross-check
Each of 70, 140, and 210 is divisible by 2, 5, and 7, while no fourth positive multiple of 70 lies within the interval. Therefore, the required number of integers is 3.