The function F(n) is defined as the product of all the conse

Question

The function F(n) is defined as the product of all the consecutive positive integers between 1 and n^2, inclusive, whereas the function G(n) is defined as the product of the squares of all the consecutive positive integers between 1 and n, inclusive. The exponent on 2 in the prime factorization of F(3)/G(3) is
=>F(3) = 1*2*3......9 = 9!
The exponent of 2 can be given by 9/2=4/2=2/2=1
=> 4+2+1=7
G(3) = 1^2*2^2*3^2

The exponent of 2 in F(3)/G(3) = 2^7/2^2 = 2^5

Hence E

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Bunuel · Accepted Answer

The function F(n) is defined as the product of all the consecutive positive integers between 1 and n^2, inclusive, thus  F(3)=1*2*3*...*9=9! .

The function G(n) is defined as the product of the  squares  of all the consecutive positive integers between 1 and n, inclusive, thus  G(3)=1^2*2^2*3^2=3!*3! .

\frac{F(3)}{G(3)}=\frac{9!}{3!*3!}=\frac{4*5*6*7*8*9}{6}=2^2*5*7*2^3*9=2^5*(5*7*9) .

The power of 2 is 5.

Answer: E.

thevenus · Answer

F(3)/G(3)
=product(1 to 3 ^2) / 1.2^2.3^2
=1.2.3.4.5.6.7.8.9/1.4.9
=1.2.3.(2^2).5.(2.3).7.(2^3).9/1.(2^2).9
=1.(2^7).3.5.7.9/1.(2^2).9
Loof for 2^7/2^2=2^5 ----Exponent 5

Answer: E

Temurkhon · Answer

Did not conceive the problem which is just to find the degree of 2 in final. So

9! factorization is

9=3*3 (remove)
8= 2*2*2 
7=7
6=3* 2 
5=5
4=2*2 (remove)
3=3
2= 2

36 factorization is
3, 4, 3 (remove)
Five 2s are bolded

it is E

PareshGmat · Answer

This problem takes more time to read than to solve

F(n) = (n^2)! F(3) = 9! ............. (1) G(n) = 1^2 * 2^2 * 3^2 .......... n^2 = (n!)^2 G(3) = (3!)^2 = 36 ............ (2) \frac{(1)}{(2)} Answer = E

Paras96 · Answer

The function F(n) is defined as the product of all the consecutive positive integers between 1 and n^2, inclusive, whereas the function G(n) is defined as the product of the squares of all the consecutive positive integers between 1 and n, inclusive. The exponent on 2 in the prime factorization of F(3)/G(3) is
=>F(3) = 1*2*3......9 = 9! 
The exponent of 2 can be given by 9/2=4/2=2/2=1
=> 4+2+1=7
G(3) = 1^2*2^2*3^2

The exponent of 2 in F(3)/G(3) = 2^7/2^2 = 2^5

Hence E

bumpbot · Answer

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