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Re: If n = 20! + 17, then n is divisible by which of the followi [#permalink]
GreenlightTestPrep wrote:
Carcass 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 III


Answer choice I: is 20! + 17 divisible by 15?
20! + 17 = (20)(19)(18)(17)(16)(15)(other stuff) + 15 + 2
= (15)(some number + 1) + 2
(15)(some number + 1) is a multiple of 15
So, (15)(some number + 1) + 2 is 2 greater than a multiple of 15
So, if we divide (15)(some number + 1) + 2 by 15, the remainder will be 2
So, 20! + 17 is NOT divisible by 15
ELIMINATE B and D

Answer choice II: is 20! + 17 divisible by 17?
20! + 17 = (20)(19)(18)(17)(other stuff) + 17
= (17)(some number + 1)
If we divide (17)(some number + 1) by 17, the remainder will be 0
So, 20! + 17 IS divisible by 17
ELIMINATE A

Answer choice III: is 20! + 17 divisible by 19?
20! + 17 = (20)(19)(other stuff) + 17
= (19)(some number) + 17
If we divide (19)(some number) + 17 by 19, the remainder will be 17
So, 20! + 17 is NOT divisible by 19
ELIMINATE E


Answer: C

Cheers,
Brent


How could we write (15)(some number + 1) + 2? Need an explanation on this.
Thank You!
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Re: If n = 20! + 17, then n is divisible by which of the followi [#permalink]
2
SivhHarish wrote:

How could we write (15)(some number + 1) + 2? Need an explanation on this.
Thank You!


We'll start with: 20! + 17

We want to determine whether this value is divisible by 15
So let's see if we can factor 15 out of 20! + 17

Since 20! = (20)(19)(18)(17)(16)(15)(more numbers), we can factor out 15 to write: 20! + 17 = 15[(20)(19)(18)(17)(16)(more numbers)]

Also recognize that 17 = 15 + 2

So.......

20! + 17 = (20)(19)(18)(17)(16)(15)(more numbers) + 15 + 2
20! + 17 = [(20)(19)(18)(17)(16)(15)(more numbers) + 15] + 2
20! + 17 = 15[(20)(19)(18)(17)(16)(more numbers) + 1] + 2

Since 15[(20)(19)(18)(17)(16)(more numbers) + 1] is clearly a multiple of 15, . . .
. . . it must be the case that 15[(20)(19)(18)(17)(16)(more numbers) + 1] + 2 is 2 greater than some multiple of 15, which means 17 is NOT divisible by 15

Does that help?
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Re: If n = 20! + 17, then n is divisible by which of the followi [#permalink]
20! is divisible by 15, 17, and 19, so 20! + 17 lands on a multiple of 17 but not 15 or 19:

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