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satish_iitg
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If the GCF of n and 16 is 4, we know n is a multiple of 4, but NOT a multiple of any higher power of 2, such as 8 or 16. (If n were a multiple of 8, for instance, the GCF of n and 16 would be 8.)
So n = 4 * (some odd integer(s)).
If the GCF and n and 45 is 3, then n is a multiple of 3, but NOT a multiple of 9 or of 5. (If it were, it would share a greater GCF with 45, which is 3 * 3 * 5.)
So n = 4 * 3 * something, where that something is NOT 5 or another 3.
Now let's go to the answers. Eliminate A (not a multiple of 2), B (not a multiple of 3), C (a multiple of 5, but n isn't a multiple of 5), and E (same reasons as C).
By process of elimination, D is our best answer.
If wanted to justify D, we could do so as follows.
210 = 2 * 3 * 5 * 7
n = 2 * 2 * 3 * (some other odd numbers)
To find the GCF, we take all common factors. n and 210 have at least two common factors (2 and 3) and CANNOT have a 5 in common, as n is not a multiple of 5. The GCF of 210 and n is thus 2 * 3 (if n is not a multiple of 7) or 2 * 3 * 7 (if n is a multiple of 7). So our answer is either 6 or 42.
