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If a is divisible by 7!, then a/5 must be

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If a is divisible by 7!, then a/5 must be [#permalink]

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New post 13 Dec 2016, 09:51
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If a is divisible by 7!, then a/5 must be

I. divisible by 6
II. a multiple of 7
III. a multiple of 10

A. I, II, and III
B. None
C. II only
D. III only
E. I and II only
[Reveal] Spoiler: OA

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Re: If a is divisible by 7!, then a/5 must be [#permalink]

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New post 13 Dec 2016, 10:36
Bunuel wrote:
If a is divisible by 7!, then a/5 must be

I. divisible by 6
II. a multiple of 7
III. a multiple of 10

A. I, II, and III
B. None
C. II only
D. III only
E. I and II only


a = 7*6*5*4*3*2*k

a/5 = 7*6*4*3*2*k

See, a/5 is divisibly by ony (I) and (II)

Answer will be (E)

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Re: If a is divisible by 7!, then a/5 must be [#permalink]

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New post 29 Nov 2017, 09:05
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Bunuel wrote:
If a is divisible by 7!, then a/5 must be

I. divisible by 6
II. a multiple of 7
III. a multiple of 10

A. I, II, and III
B. None
C. II only
D. III only
E. I and II only


-----ASIDE---------------------
RULE: If integer N is divisible by integer d, we can write N = dk (for some integer k)
Example: 30 is divisible by 6, and we can write 30 = (6)(5).
Likewise, 100 is divisible by 4, and we can write 100 = (4)(25).
Or, if q is divisible by 5, and we can write q = 5k (where k is some integer).
-----ONTO THE QUESTION!---------------------

Given: a is divisible by 7!
So, a = (7!)(k) for some integer k
That is, a = (7)(6)(5)(4)(3)(2)(1)(k)

This means: a/5 = (7)(6)(5)(4)(3)(2)(1)(k)/5
Simplify to get: a/5 = (7)(6)(4)(3)(2)(1)(k)

Now let's examine the statements:
I. divisible by 6
a/5 = (7)(6)(4)(3)(2)(1)(k)
So, a/5 MUST be divisible by 6
Statement I is true

II. a multiple of 7
a/5 = (7)(6)(4)(3)(2)(1)(k)
So, a/5 MUST be divisible by 7
In other words, a/5 MUST be a multiple of 7
Statement II is true

III. a multiple of 10
a/5 = (7)(6)(4)(3)(2)(1)(k)
If k = 1, then a/5 is definitely NOT a multiple of 10, since we cannot see a 10 "hiding" in the prime factorization of a/5
Statement III need NOT be true

Answer: E

-----ASIDE---------------------
Here's more information about the concept of divisors "hiding" in the prime factorization of a number:

A lot of integer property questions can be solved using prime factorization.
For questions involving divisibility, divisors, factors and multiples, we can say:
If N is divisible by k, then k is "hiding" within the prime factorization of N

Consider these examples:
24 is divisible by 3 because 24 = (2)(2)(2)(3)
Likewise, 70 is divisible by 5 because 70 = (2)(5)(7)
And 112 is divisible by 8 because 112 = (2)(2)(2)(2)(7)
And 630 is divisible by 15 because 630 = (2)(3)(3)(5)(7)

EVEN MORE HERE:

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Kudos [?]: 2758 [2], given: 364

Expert Post
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Joined: 02 Sep 2009
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Kudos [?]: 135771 [0], given: 12708

Re: If a is divisible by 7!, then a/5 must be [#permalink]

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New post 03 Dec 2017, 07:22
Bunuel wrote:
If a is divisible by 7!, then a/5 must be

I. divisible by 6
II. a multiple of 7
III. a multiple of 10

A. I, II, and III
B. None
C. II only
D. III only
E. I and II only


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https://gmatclub.com/forum/if-a-is-divi ... 29901.html
https://gmatclub.com/forum/if-a-is-divi ... 28285.html
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Kudos [?]: 135771 [0], given: 12708

Re: If a is divisible by 7!, then a/5 must be   [#permalink] 03 Dec 2017, 07:22
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