Divisibility

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Uva@90: Master | Next Rank: 500 Posts; Posts: 490; Joined: Thu Jul 04, 2013 7:30 amFollowed by: 5 members; Location: Chennai, India; Thanked: 83 times

by Uva@90 » Wed Oct 09, 2013 7:26 pm

Hi All,
Please help me with the below problem,

Y= (3^5 - 3^2)^2/(5^7 - 5^4)^-2, then Y is not divisible by which of the following?

A. 6^4
B. 62^2
C. 65^2
D. 15^2
E. 52^4

OA: E

Thanks in advance.

Regards,
Uva.

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Source: Beat The GMAT — Problem Solving | Wed Oct 09, 2013 7:26 pm

ganeshrkamath: Master | Next Rank: 500 Posts; Posts: 283; Joined: Sun Jun 23, 2013 11:56 pmFollowed by: 26 members; Location: Bangalore, India; Thanked: 97 times; GMAT Score:750

by ganeshrkamath » Wed Oct 09, 2013 7:53 pm

Uva@90 wrote:Hi All,
Please help me with the below problem,

Y= (3^5 - 3^2)^2/(5^7 - 5^4)^-2, then Y is not divisible by which of the following?

A. 6^4
B. 62^2
C. 65^2
D. 15^2
E. 52^4

OA: E

Thanks in advance.

Regards,
Uva.

3^5 - 3^2 = 3^2 * (3^3 - 1) = 3^2 * (26) = 3^2 * 2 * 13

5^7 - 5^4 = 5^4 * (5^3 - 1) = 5^4 * (124) = 5^4 * 4 * 31

Y = (3^5 - 3^2)^2/(5^7 - 5^4)^-2
= (3^2 * 2 * 13 * 5^4 * 4 * 31)^2
= 2^6 * 3^4 * 5^8 * 13^2 * 31^2

For a number to evenly divide Y, it should:
1. Not have any factors that are not factors of Y
2. Not have power of any factor greater than the power of the same factor in Y

Option A = 6^4 = 2^4 * 3^4 => Divides Y
Option B = 62^2 = 2^2 * 31^2 => Divides Y
Option C = 65^2 = 5^2 * 13^2 => Divides Y
Option D = 15^2 = 3^2 * 5^2 => Divides Y
Option E = 52^4 = 4^4 * 13^4 = 2^8 * 13^4 => Does NOT divide Y (Fails condition 2)

Choose E

Cheers

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Uva@90: Master | Next Rank: 500 Posts; Posts: 490; Joined: Thu Jul 04, 2013 7:30 amFollowed by: 5 members; Location: Chennai, India; Thanked: 83 times

by Uva@90 » Thu Oct 10, 2013 3:44 am

ganeshrkamath wrote:
Uva@90 wrote:Hi All,
Please help me with the below problem,

Y= (3^5 - 3^2)^2/(5^7 - 5^4)^-2, then Y is not divisible by which of the following?

A. 6^4
B. 62^2
C. 65^2
D. 15^2
E. 52^4

OA: E

Thanks in advance.

Regards,
Uva.
3^5 - 3^2 = 3^2 * (3^3 - 1) = 3^2 * (26) = 3^2 * 2 * 13

5^7 - 5^4 = 5^4 * (5^3 - 1) = 5^4 * (124) = 5^4 * 4 * 31

Y = (3^5 - 3^2)^2/(5^7 - 5^4)^-2
= (3^2 * 2 * 13 * 5^4 * 4 * 31)^2
= 2^6 * 3^4 * 5^8 * 13^2 * 31^2

For a number to evenly divide Y, it should:
1. Not have any factors that are not factors of Y
2. Not have power of any factor greater than the power of the same factor in Y

Option A = 6^4 = 2^4 * 3^4 => Divides Y
Option B = 62^2 = 2^2 * 31^2 => Divides Y
Option C = 65^2 = 5^2 * 13^2 => Divides Y
Option D = 15^2 = 3^2 * 5^2 => Divides Y
Option E = 52^4 = 4^4 * 13^4 = 2^8 * 13^4 => Does NOT divide Y (Fails condition 2)

Choose E

Cheers

Thanks Ganesh, your solution is soo clear.

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