If x and y are integers, which of the following CANNOT be th

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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Thu Jan 25, 2018 3:00 am

[GMAT math practice question]

If x and y are integers, which of the following CANNOT be the value of x^2+y^2?

A. 121
B. 122
C. 123
D. 125
E. 130

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DrMaths: Senior | Next Rank: 100 Posts; Posts: 82; Joined: Mon Jan 15, 2018 2:01 am

by DrMaths » Thu Jan 25, 2018 7:41 am

The smallest Pythagorean triples are:
(3, 4, 5) (5, 12, 13) (8, 15, 17) (7, 24, 25)
None of these include 11
Answer A is 11^2 = 121

As 11 is not in the list of triples, then A is impossible.
All the other values are not squares, so non-integer solutions are possible.

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Thu Jan 25, 2018 9:27 am

DrMaths wrote:The smallest Pythagorean triples are:
(3, 4, 5) (5, 12, 13) (8, 15, 17) (7, 24, 25)
None of these include 11
Answer A is 11^2 = 121

As 11 is not in the list of triples, then A is impossible.
All the other values are not squares, so non-integer solutions are possible.

'

Be careful; if x = 0 and y = 11, then we get: 0Â² + 11Â² = 121
Eliminate A

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Thu Jan 25, 2018 9:32 am

Max@Math Revolution wrote:[GMAT math practice question]

If x and y are integers, which of the following CANNOT be the value of x^2+y^2?

A. 121
B. 122
C. 123
D. 125
E. 130

Let's do this systematically:
0Â² = 0
1Â² = 1
2Â² = 4
3Â² = 9
4Â² = 16
5Â² = 25
6Â² = 36
7Â² = 49
8Â² = 64
9Â² = 81
10Â² = 100
11Â² = 121

A. 121 = 0Â² + 11Â² ELIMINATE A
B. 122 = 1Â² + 11Â² ELIMINATE B
C. 123 = can't eliminate
D. 125 = 10Â² + 5Â² ELIMINATE D
E. 130 = 9Â² + 7Â² ELIMINATE E

Answer: C

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Thu Jan 25, 2018 11:02 am

Max@Math Revolution wrote:[GMAT math practice question]

If x and y are integers, which of the following CANNOT be the value of x^2+y^2?

A. 121
B. 122
C. 123
D. 125
E. 130

Eliminate the four answer choices that CAN be the value of xÂ²+yÂ², where x and y are integers.
A: 121 = 11Â² + 0Â²
B: 122 = 11Â² + 1Â²
D: 125 = 11Â² + 2Â²
E: 130 = 11Â² + 3Â²
Eliminate A, B, D and E.

The correct answer is C.

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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Sun Jan 28, 2018 5:39 pm

=>
Squares of even integers, (2k)^2 = 4k^2 have the remainder 0, when they are divided by 4.

Squares of odd integers, (2k+1)^2 = 4k^2 + 4k + 1 have the remainder 1, when they are divided by 4.

Hence, squares of integers can have remainders of 0 or 1 only, when they are divided by 4. So, the sum of two squares of integers cannot have the remainder of 3 when it is divided by 4.

Thus, 123 cannot be the value of x^2+y^2.
Therefore, the answer is C.
Answer: C

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Jeff@TargetTestPrep: GMAT Instructor; Posts: 1462; Joined: Thu Apr 09, 2015 9:34 amFollowed by: 22 members; Location: New York, NY; Thanked: 39 times

by Jeff@TargetTestPrep » Mon Jan 29, 2018 9:42 am

Max@Math Revolution wrote:[GMAT math practice question]

If x and y are integers, which of the following CANNOT be the value of x^2+y^2?

A. 121
B. 122
C. 123
D. 125
E. 130

Let's test our answer choices:

A) 121

121 = 0^2 + 11^2

B) 122

122 = 1^1 + 11^2

C) 123

There are no two perfect squares that sum to 123.

Answer: C

Jeffrey Miller
Head of GMAT Instruction
[email protected]

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