kiran120680 wrote:
If positive integer x is a multiple of 14 and 12, which of the following statements must be true?
I. x is divisible by 8
II. x is divisible by 28
III. The greatest common divisor of x and 9 is 3
A. I only
B. II only
C. III only
D. I and II only
E. II and III only
We recall that if a number is a multiple of two numbers, it’s also a multiple of the LCM of the two numbers. The LCM of 14 and 12 is 7 * 4 * 3 = 84. So x is a multiple of 84.
We also recall that if a is a multiple of b, then it’s also a multiple of any factors of b.
Thus, we see that x MUST BE divisible by 28 (since 28 is a factor of 84) but not necessarily by 8 (since 8 is not a factor 84). Thus, we see that statement I is not necessarily true, and statement II must be true.
The GCF of x and 9 does not have to be 3. For example, if x = 84 * 3 = 252, then the GCF of x and 9 is 9. Thus, statement III is not necessarily true.
Answer: B
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