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What is the sum of the remainders when the first 30 positive

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by Max@Math Revolution » Mon Jun 18, 2018 11:56 pm

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[GMAT math practice question]

What is the sum of the remainders when the first 30 positive integers are divided by 5?

A. 50
B. 55
C. 60
D. 65
E. 70
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by arosman » Wed Jun 20, 2018 1:43 pm
Remainder problems often have a pattern to them. Name of the game is then find the pattern.

1/5 = 0R1
2/5 = 0R2
3/5 = 0R3
4/5 = 0R4
5/5 = 1R0
6/5 = 1R1
7/5 = 1R2
Etc.....................

Each cycle has a sum of 0+1+2+3+4 = 10. We have 6 cycles ---------> 10 * 6 = 60

Answer: C
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by Scott@TargetTestPrep » Wed Jun 20, 2018 4:25 pm
Max@Math Revolution wrote:[GMAT math practice question]

What is the sum of the remainders when the first 30 positive integers are divided by 5?

A. 50
B. 55
C. 60
D. 65
E. 70
1/5 = 0 R 1, 2/5 = 0 R 2, 3/5 = 0 R 3, 4/5 = 0 R 4, 5/5 = 1 R 0

We see that when the first 5 positive integers are divided by 5, the remainders are 1, 2, 3, 4, and 0, respectively. So the sum of these remainders is 10.

When the next 5 positive integers (6 to 10, inclusive) are divided by 5, the remainders will also be 1, 2, 3, 4, and 0, respectively. So the sum of the remainders will be 10 also. This is true for every set of 5 consecutive integers: 5n - 4, 5n - 3, 5n - 2, 5n - 1, 5n where n is a positive integer.

We see that the first 30 positive integers are comprised of 6 sets of 5 integers, and each set produces 10 as the sum of the remainders. Thus, the sum of all the remainders when the first 30 positive integers are divided by 5 is 6 x 10 = 60.

Answer: C

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by Max@Math Revolution » Thu Jun 21, 2018 12:33 am
=>

1, 6, 11, 16, 21 and 26 have remainder 1 when they are divided by 5.
2, 7, 12, 17, 22 and 27 have remainder 2 when they are divided by 5.
3, 8, 13, 18, 23 and 28 have remainder 3 when they are divided by 5.
4, 9, 14, 19, 24 and 29 have remainder 4 when they are divided by 5.
5, 10, 15, 20, 25 and 30 have remainder 0 when they are divided by 5.

The sum of the remainders is
1*6 + 2*6 + 3*6 + 4*6 + 0*6 = ( 1 + 2 + 3 + 4 + 0 ) * 6 = 60.

Therefore, the answer is C.

Answer: C
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