Bunuel wrote:
If the greatest common factor of positive integers n and m is 15, and the remainder when \((n+x)^{32}\) is divided by 15 is 1, which of the following CANNOT be the value of x?
A. 1
B. 4
C. 9
D. 11
E. 14
Are You Up For the Challenge: 700 Level QuestionsSince the GCF of n and m is 15, n must be a multiple of 15. If we expand (n + x)^32, every term in the expansion will have a factor of n except the last term, which is x^32. The terms that have a factor of n will be divisible by 15 (thus, the remainder is 0 when they are divided by 15). Therefore, the only way the remainder could be 1 when (n + x)^32 is divided by 15 would be if x^32 divided by 15 would yield a remainder of 1.
Now, let’s look at the choices.
A) 1
If x = 1, then 1^32 = 1 and when 1 is divided by 15, the remainder is 1.
B) 4
If x = 4, then 4^32 = (4^2)^16 = 16^16. Notice that when 16 is divided by 15, the remainder is 1. Therefore, when 16^16 is divided by 15, the remainder will be same as when 1^16 is divided by 15, and therefore, that remainder will be 1.
Let’s skip C for the moment and look at D and E first.
D) 11
Since 11 - 15 = -4, then x = 11 is equivalent to x = -4. However, since (-4)^32 = 4^32, then the remainder will be equal to the remainder when x = 4, which is 1.
E) 14
Similarly, since 14 - 15 = -1, then x = 14 is equivalent to x = -1. However, since (-1)^32 = 1^32, then the remainder will be equal to the remainder when x = 1, which is 1.
We have rejected choices A, B, D, and E. Therefore, the correct answer must be C.
Answer: C
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