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# Which of the following MUST yield an integer when divided by 5?

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Math Expert
Joined: 02 Sep 2009
Posts: 60727
Which of the following MUST yield an integer when divided by 5?  [#permalink]

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15 Jul 2015, 03:20
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15% (low)

Question Stats:

84% (01:11) correct 16% (01:26) wrong based on 115 sessions

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Which of the following MUST yield an integer when divided by 5?

A. The sum of five consecutive positive integers.
B. The square of a prime number.
C. The sum of two odd integers.
D. The product of three consecutive odd numbers.
E. The difference between a multiple of 8 and a multiple of 3.

Kudos for a correct solution.

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Re: Which of the following MUST yield an integer when divided by 5?  [#permalink]

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15 Jul 2015, 04:02
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A . let numbers be x-2, x-1, x, x+1, x+2; their sum 5x --> Always Divisible
Problems wither options

B. Prime number can be 3,7.

C. again 3+9 = 12

D. 3 consecutive odd integers may be = 17,19,21 (Not divisible by 5)

E. Multiple of 8 : 8x, Multiple of 3 : 3y; Their Difference 8x-3y (Not Necessarily divisible by 5, Only possible if x=y)
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Re: Which of the following MUST yield an integer when divided by 5?  [#permalink]

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15 Jul 2015, 05:50
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For option A: Let the numbers be (x-2), (x-1), (x), (x+1), (x+2).

Now, (x-2)+(x-1) + (x) + (x+1) + (x+2) = 5(x). Hence A is true. No need to check other answers since there can only be one correct answer.

So, Ans(A).
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Re: Which of the following MUST yield an integer when divided by 5?  [#permalink]

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15 Jul 2015, 04:47
Bunuel wrote:
Which of the following MUST yield an integer when divided by 5?

A. The sum of five consecutive positive integers.
B. The square of a prime number.
C. The sum of two odd integers.
D. The product of three consecutive odd numbers.
E. The difference between a multiple of 8 and a multiple of 3.

Kudos for a correct solution.

Straightforward question:

As this is a MUST be true question, the correct answer MUST stay true for all possible values.

A can be proven as follows : The 5 consecutive integers can be represenetd as : x, x+1, x+2, x+3, x+4 (x can be any positive integer value). Thus their sum = 5x+10 =5(x+2)---> Thus this sum is ALWAYS divisible by 5 and is thus the correct answer.

For proving other statements as false:

B : 2^2 is not divible by 5.
C: 3+11 is not divisble by 5
D: 7*9*11 is not divisible by 5
E: 24-3 is not divisble by 5
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Joined: 20 Jul 2011
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Re: Which of the following MUST yield an integer when divided by 5?  [#permalink]

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17 Jul 2015, 22:39
Bunuel wrote:
Which of the following MUST yield an integer when divided by 5?

A. The sum of five consecutive positive integers.
B. The square of a prime number.
C. The sum of two odd integers.
D. The product of three consecutive odd numbers.
E. The difference between a multiple of 8 and a multiple of 3.

Kudos for a correct solution.

A. Plugin numbers: 1+2+3+4+5= 15/5, 12+13+14+15+16=70/5, Hence this holds true for all cases. Lets check the other options as well.
B. 2^2 = 4/5(Not divisible), 5^2(divisible). Not always true.
C. 5+3 not divisible, but 3 + 7 is divisible. Not always true.
D. 1*3*5 is divisible, but 7*9*11 is not. Not always true.
E. 8-3 is divisible, but 16-3 is not.

Math Expert
Joined: 02 Sep 2009
Posts: 60727
Re: Which of the following MUST yield an integer when divided by 5?  [#permalink]

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19 Jul 2015, 13:12
Bunuel wrote:
Which of the following MUST yield an integer when divided by 5?

A. The sum of five consecutive positive integers.
B. The square of a prime number.
C. The sum of two odd integers.
D. The product of three consecutive odd numbers.
E. The difference between a multiple of 8 and a multiple of 3.

Kudos for a correct solution.

800score Official Solution:

(A) Another way to phrase the question is: Which of the following is divisible by 5?

Choice (A) might take some time to figure out. Others are easier.

(B) is wrong — many squares of primes, such as 2², are not divisible by 5.

(C) is also wrong: consider 3 + 9 = 12.

(D) is incorrect; one example is (7)(9)(11), which is not divisible by 5.

(E) can be shown to be wrong as well: For instance, the difference between 16 and 9 is 7.

Choice (A), then, must be correct. If the first of the five consecutive positive integers is x, the full list is:
x + (x +1) + (x + 2) + (x + 3) + (x + 4) = 5x + 10

Since x is an integer, 5x is divisible by 5. 10 is also divisible by 5. Then, 5x + 10 is divisible by 5.
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Re: Which of the following MUST yield an integer when divided by 5?   [#permalink] 19 Jul 2015, 13:12
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