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If n is a three-digit prime number and j is an integer, which of the following is NOT a possible value of k, where k is the smallest positive integer such that n – 5j = k?
Rearrange the equation for k to isolate n:
n = 5j + k
You are told that k is to be the “smallest positive integer” that makes this equation true for some three-digit prime number n, given that j is an integer.
This equation may remind you of remainders (especially if you glance at the answer choices). Since j is an integer, the term 5j just means “multiple of 5.” The integer k is defined to be positive, so instead of 0 as the remainder, you have 5 as its replacement. So you’re “sort of” being asked this: which of the following numbers CANNOT be the remainder after you divide a three-digit prime number by 5?
Well, a three-digit prime number cannot be a multiple of 5. The only prime number that is a multiple of 5 is 5 itself, which only has one digit. So n cannot be written as 5j + 5. Every other remainder is a possibility: 101, 103, 107, and 109 in fact are all primes, and they give you 1, 3, 2, and 4 as possible values of k.
The correct answer is E.
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