If 6^y is a factor of (10!)^2, What is the greatest possible

Question

If 6^y is a factor of (10!)^2, What is the greatest possible value of y ?

A. 2
B. 4
C. 6
D. 8
E. 10

Bunuel · Accepted Answer

6=2*3. Now, there will be obviously less 3's than 2's in (10!)^2, so maximum power of 3 will be limiting factor for maximum power of 6, which is y. Finding the maximum powers of a prime number 3, in 10!: \frac{10}{3}+\frac{10}{3^2}=3+1=4 (take only quotients into account). So, we have that the maximum power of 3 in 10! is 4, thus maximum power of 3 in (10!)^2 will be 8: (3^4)^2=3^8 . As discussed 8 is the maximum power of 6 as well. Answer: D. For more on this subject check: everything-about-factorials-on-the-gmat-85592.html (explanation of this concept in details). Similar questions to practice: p-and-q-are-integers-if-p-is-divisible-by-10-q-and-cannot-109038.html question-about-p-prime-in-to-n-factorial-108086.html how-many-zeros-does-100-end-with-100599.html if-n-is-the-product-of-integers-from-1-to-20-inclusive-106289.html what-is-the-greatest-value-of-m-such-that-4-m-is-a-factor-of-105746.html find-the-number-of-trailing-zeros-in-the-product-of-108248.html find-the-number-of-trailing-zeros-in-the-expansion-of-108249.html if-d-is-a-positive-integer-and-f-is-the-product-of-the-first-126692.html if-m-is-the-product-of-all-integers-from-1-to-40-inclusive-108971.html if-10-2-5-2-is-divisible-by-10-n-what-is-the-greatest-106060.html Hope it helps.

vomhorizon · Answer

Factorial of 10 can be written as 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

or 2 x 5 x 3 x 3 x 2 x 2 x 2 x 7 x 2 x 3 x 5 x 2 x 2 x 3 x 2 x 1

So in 10 factorial we have 08 2's and 4 three's ... Square of 10 factorial will give us 16 2's and 8 three's

to get six we know that we would have to take one 2 and one three ..the maximum number of three's we can take is 8 therefore 8 different 6's can be formed therefore max possible value of y can be 6 ..

2 x 3
2 x 3 
2 x 3
2 x 3
2 x 3
2 x 3
2 x 3
2 x 3

D..

pappueshwar · Answer

hi bunuel,

i did nt understand how did u get 3's more than 2's in 10!.

i am of the view that 10! = 10*9*8*7*6*5*4*3*2*1

if we expand this we get =  2*5   *3*3   *2*2*2  * 7  * 3*2 * 5*  2*2 *  3 *  2  *1

so there are 8 2's and 4 3's in the expansion above.

so how to interpret this and proceed

Bunuel · Answer

It's: "there will be obviously LESS 3's than 2's in (10!)^2, so  maximum power of 3 will be limiting factor  for maximum power of 6, which is y."

essarr · Answer

wow, that link really helped thanks soooo much; it's so much simpler now that I understand the concept

hellscream · Answer

Thank Bunuel for very clear and concise answer.

PareshGmat · Answer

(10!)^2 = 10^2 *  9^2  * 8^2 * 7^2 *  6^2  * 5^2 * 4^2 *  3^2  * 2^2

Just concentrate on the power of 3 (Power of 2's would be more as compared to 3; so it can be ignored)

9^2  =  3^4

6^2  =  3^2  * 2^2

3^2  =  3^2

Total powers of 3 = 8

Answer = 8

chetan2u · Answer

Hi,

Since you already know there are 8 3s and 14 2s, I'll start from thereon...
Each 6 is composed of 3 and 2...
So it requires equal number of 3 and 2..

But (14-8) that is 6 of 2s do not have a 3 to make a 6..
Therefore the number of 6s will depend on the integer (3or2) with lowest value thus answer is 8..

EgmatQuantExpert · Answer

Hey, PFB the solution. • 6^y can be written as 2^y * 3^y • To find the greatest possible value of y, we need to find out how many 3 's are there in (10!)^2 • Now 10! = 1 * 2 *3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 • Which can be written as - o 10! = 2^8 * 3^4 * 5^2 * 7^1 • Therefore (10!)^2 = 2^{16} * 3^8 * 5^4 * 7^2 • As we can see there are 16 2's but only 8 3's • But to make a 6 we need both one 2 and one 3 . • Therefore, the maximum number of 6 's that we can make is 8 . • Please note : that out of the 16 2's we can use only 8 of them and the rest 8 cannot be clubbed with any 3's, as there aren't any left. • Hence, the value of y = 8 . Thanks, Saquib Quant Expert e-GMAT Register for our Free Session on Number Properties (held every 3rd week) to solve exciting 700+ Level Questions in a classroom environment under the real-time guidance of our Experts

https://e-gmat.com/blogs/wp-content/uploads/2016/12/NP-Free-Trial-e1481107541356.png

Akriti_Khetawat · Answer

Hey what's a limiting factor and why are we considering only that number to find the answer?

Bunuel · Answer

There will be obviously less 3's than 2's in (10!)^2, so the power of 3 in (10!)^2 will determine the power of 6 in (10!)^2. Please follow the links given in that post for more.

bumpbot · Answer

Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.

If 6^y is a factor of (10!)^2, What is the greatest possible

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

GMAT Problem Solving (PS) Questions

If 6^y is a factor of (10!)^2, What is the greatest possible

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia