If x is the smallest positive integer that is divisible by each of the

Question

If x is the smallest positive integer that is divisible by each of the integers from 24 to 27 inclusive, which of the following statements must be true?

I. x > 3200
II. x is a multiple of 90
III. 500 is a factor of x

(A) I only
(B) II only
(C) I and II
(D) II and III
(E) I,II,III

Kinshook · Answer

Given; If x is the smallest positive integer that is divisible by each of the integers from 24 to 27 inclusive, which of the following statements must be true?

x is the smallest positive integer divisible by 24=2^3*3, 25=5^2, 26=2*13 & 27=3^3
x = 2^3*3^3*5^2*13 = 8*27*25*13 = 70200

I. x > 3200 : TRUE
II. x is a multiple of 90; 90 = 2*3^2*5; TRUE
III. 500 is a factor of x; FALSE

IMO C

Vatsal7794 · Answer

So LCM of 24 25 36 and 27 = 2^3 * 3^3 * 5^2 * 13

1) Condition satisfies
2)Again condition satisfies
3) Split 500 into prime factors - 5^3 * 2^2

Value of X has two 5"S but 500 has three 5's Therefore 500 is not a factor of X
And answer is C

Please comment if you there is anything wrong or if you faced any problem with my solution

KarishmaB · Answer

We need the LCM of 24, 25, 26 and 27
24 = 3 * 2*2*2
25 = 5*5
26 = 2 * 13
27 = 3*3*3

LCM = 2*2*2 * 3*3*3 * 5*5 * 13

I. x > 3200

3200 = 5*5 * 128
Other than 5*5, LCM has 8*9*13 which is definitely greater than 128.

II. x is a multiple of 90

90 = 2 * 2*2 * 5
All of these factors are factors of the LCM so LCM is divisible by 90.

III. 500 is a factor of x

500 = 2*2 * 5*5*5
LCM has only two 5s while 500 has three 5s.

Answer (C)

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If x is the smallest positive integer that is divisible by each of the

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