swerve 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 III
E. II and III

The OA C

Source: Official Guide

Since 20! is a multiple of 17 and 17 is a multiple of 17, we know that 20! + 17 is divisible by 17. On the other hand, since 20! is a multiple of 15 but 17 is not, then 20! + 15 is not divisible by 15. Likewise, 20! + 17 is not divisible by 19, either.

Answer: C

Hi All,

We're told that N = (20! + 17). We're asked which of the following is N divisible by. This question is ultimately about "factoring" and why numbers divide evenly into other numbers. I'm going to start with a simple example and work up to the details in this prompt. You probably know that 3 divides evenly into 3! (3! = 1x2x3). We can factor out a 3 and get 3(2); mathematically, this means that 3 divides evenly into 3!

Does 3 divide into 3! + 1? Does 3 divide into 7? No, because you CAN'T factor out a 3.
Does 3 divide into 3! + 2? Does 3 divide into 8? No, because you CAN'T factor out a 3.
Does 3 divide into 3! + 3? Does 3 divide into 9? YES, because you CAN factor out a 3. You'd have 3(1x2 + 1).

This same rule applies to this much larger value: 20! + 17

We can factor out a 17, which would give us (17)(20x19x18x16x15....x3x2x1 + 1). This tells us that N is absolutely divisibly by 17. That extra "+1" in the calculation means that NONE of the other numbers from 1 to 20 will divide into 20! + 17 though (since they cannot be factored out). Thus, the only one of the 3 Roman Numerals that divides evenly in is 17.

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

swerve 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 III
E. II and III

The OA C

Source: Official Guide

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
So, if we divide (15)(some number + 1) + 2 by 15, the remainder will be 2
So, 20! + 17 is NOT divisible by 15


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


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

Answer: C

Cheers,
Brent