Walkabout wrote:
If n = 20! + 17, then n is divisible by which of the following?
I. 15
II. 17
III. 19
(A) None
(B) I only
(C) II only
(D) I and II
(E) II and II
We are given that n = 20! + 17 and need to know whether n is divisible by 15, 17, and/or 19. To determine this, we rewrite the given expression for n using each answer choice.
Thus, we have:
Does (20! + 17)/15 = integer?
Does (20! + 17)/17 = integer?
Does (20! + 17)/19 = integer?
We now use the distributive property of division over addition to determine which of these expressions is/are equal to an integer.
The distributive property of division over addition tells us that (a + c)/b = a/b + c/b. We apply this rule as follows:
I.
Does (20! + 17)/15 = integer?
Does 20!/15 + 17/15 = integer?
Although 20! is divisible by 15, 17 is NOT, and thus (20! + 17)/15 IS NOT an integer.
We can eliminate answer choices B and D.
II.
Does (20! + 17)/17 = integer?
Does 20!/17 + 17/17 = integer?
Both 20! and 17 are divisible by 17, and thus (20! + 17)/17 IS an integer.
We can eliminate answer choice A.
III.
Does (20! + 17)/19 = integer?
Does 20!/19 + 17/19 = integer?
Although 20! is divisible by 19, 17 is NOT, so (20! + 17)/19 IS NOT an integer.
We can eliminate answer choice E.
Thus, II is the only correct statement.
Answer: C
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