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Which of the following CANNOT be expressed as the sum of squares of tw

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Math Expert
Joined: 02 Sep 2009
Posts: 50621
Which of the following CANNOT be expressed as the sum of squares of tw  [#permalink]

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12 Jul 2018, 00:16
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5% (low)

Question Stats:

91% (01:09) correct 9% (01:09) wrong based on 35 sessions

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Which of the following CANNOT be expressed as the sum of squares of two integer?

A 13
B 17
C 21
D 29
E 34

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Joined: 07 Dec 2017
Posts: 799
Re: Which of the following CANNOT be expressed as the sum of squares of tw  [#permalink]

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12 Jul 2018, 00:54
Bunuel wrote:
Which of the following CANNOT be expressed as the sum of squares of two integer?

A 13
B 17
C 21
D 29
E 34

As we don't have that many options, we'll test them all.
This is an Alternative approach.

We only need to look at the squares up to 5^2 (as 6^2 = 36 is larger than our largest value).
So there are a total of 5 choose 2 = 10 options.
1 + 4 = 5
1 + 9 = 10
1 + 16 = 17 **
1 + 25 = 26
4 + 9 = 13 **
4 + 16 = 20
4 + 25 = 29 **
9 + 16 = 25
9 + 25 = 34 **
16 + 25 = 41

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Director
Joined: 20 Feb 2015
Posts: 796
Concentration: Strategy, General Management
Re: Which of the following CANNOT be expressed as the sum of squares of tw  [#permalink]

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12 Jul 2018, 01:30
Bunuel wrote:
Which of the following CANNOT be expressed as the sum of squares of two integer?

A 13
B 17
C 21
D 29
E 34

13=2^2+3^2
17=4^2+1^2
21=Cannot be expressed
29=5^2+2^2
34=5^2+3^2

C
Director
Joined: 31 Oct 2013
Posts: 786
Concentration: Accounting, Finance
GPA: 3.68
WE: Analyst (Accounting)
Re: Which of the following CANNOT be expressed as the sum of squares of tw  [#permalink]

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12 Jul 2018, 02:53
Bunuel wrote:
Which of the following CANNOT be expressed as the sum of squares of two integer?

A 13
B 17
C 21
D 29
E 34

A bit tricky question. Number testing is the way to get the answer.

$$3^2$$ + $$2^2$$ = 13
$$4^2$$ + $$1^1$$ = 17
$$5^2$$+ $$2^2$$ = 29
$$5^2$$ + $$3^2$$ = 34

Now , option C is left. That's our answer. Try different number less than 21. There is a reason. 21 = 3*7 or 21*1. See none of the factors are square of an integer except 1. Thus it is also impossible to express 21 as a sum of the square of the 2 integers.

Director
Status: Learning stage
Joined: 01 Oct 2017
Posts: 930
WE: Supply Chain Management (Energy and Utilities)
Re: Which of the following CANNOT be expressed as the sum of squares of tw  [#permalink]

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12 Jul 2018, 12:59
1
Bunuel wrote:
Which of the following CANNOT be expressed as the sum of squares of two integer?

A 13
B 17
C 21
D 29
E 34

Every prime of the form (4k+1) can be expressed as the sum of two squares..
Among the answer options 13,17, and 29 are prime and they can be written in the form (4k+1). So, eliminate A,B, and D.

when the number is not prime:- If each of the factors of the integer can be written as the sum of two squares is itself expressible as the sum of two squares.
Prime factorization of 34=2*17, ($$2=1^2+1^2$$ & 17 is in the form of 4k+1. So, both 2 & 17 are perfect squares)
Therefore, 34 is a sum of squares of two integer.So, eliminate E.

Ans. (C)
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Re: Which of the following CANNOT be expressed as the sum of squares of tw &nbs [#permalink] 12 Jul 2018, 12:59
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