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# What is the sum of the remainders when the first 50 positive integers

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Math Revolution GMAT Instructor
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What is the sum of the remainders when the first 50 positive integers [#permalink]

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09 Nov 2017, 01:03
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Difficulty:

5% (low)

Question Stats:

80% (00:51) correct 20% (00:43) wrong based on 59 sessions

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[GMAT math practice question]

What is the sum of the remainders when the first 50 positive integers are divided by 5?

A. 60
B. 70
C. 80
D. 90
E. 100
[Reveal] Spoiler: OA

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What is the sum of the remainders when the first 50 positive integers [#permalink]

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09 Nov 2017, 02:33
MathRevolution wrote:
[GMAT math practice question]

What is the sum of the remainders when the first 50 positive integers are divided by 5?

A. 60
B. 70
C. 80
D. 90
E. 100

$$50$$ positive integers can be visualized as $$10$$ sets of $$5$$ elements each.

each of the elements when divided by $$5$$ will have remainders as {$$1,2,3,4,0$$}. Sum of each elements of this set $$=10$$ and we have $$10$$ such sets.

so total sum of remainders $$= 10*10 = 100$$

Option E

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What is the sum of the remainders when the first 50 positive integers [#permalink]

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12 Nov 2017, 08:36
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MathRevolution wrote:
[GMAT math practice question]

What is the sum of the remainders when the first 50 positive integers are divided by 5?

A. 60
B. 70
C. 80
D. 90
E. 100

Remainders, first 10 integers, including 1-5:

$$\frac{1}{5}$$ =0 remainder 1, $$\frac{2}{5}$$ =0 remainder 2

$$\frac{3}{5}$$ =0 remainder 3, $$\frac{4}{5}$$=0 remainder 4

$$\frac{5}{5}$$ = 1 remainder 0,$$\frac{6}{5}$$ = 1 remainder 1

$$\frac{7}{5}$$ =1 remainder 2, $$\frac{8}{5}$$ =1 remainder 3

$$\frac{9}{5}$$ =1 remainder 4, $$\frac{10}{5}$$ = 2 remainder 0

For integers 1-10, sum of remainders = 1+2+3+4+1+2+3+4 = 20. That pattern holds:

1 -10: sum = 20
11-20: sum = 20
21-30: sum = 20
31-40: sum = 20
41-50: sum = 20

Total= 100

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Re: What is the sum of the remainders when the first 50 positive integers [#permalink]

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12 Nov 2017, 08:44
MathRevolution wrote:

What is the sum of the remainders when the first 50 positive integers are divided by 5?

A. 60
B. 70
C. 80
D. 90
E. 100

Let's see if we can find a pattern:

The remainders when 1, 2, 3, 4, and 5 are divided by 5 are 1, 2, 3, 4, and 0, respectively.

We get the same results from numbers 6 to 10, 11 to 15, etc.

Thus, the sum of remainders of each set of five numbers is 1 + 2 + 3 + 4 + 0 = 10.

Since there are 10 sets of five numbers from 1 to 50 inclusive, the sum of all the remainders is 10 x 10 = 100.

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Math Revolution GMAT Instructor
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Re: What is the sum of the remainders when the first 50 positive integers [#permalink]

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12 Nov 2017, 19:02
=>

1, 2, 3, 4, 5 have remainders 1, 2, 3, 4, 0 and the sum of their remainders is 1 + 2 + 3 + 4 + 0 = 10.
6, 7, 8, 9, 10 have remainders 1, 2, 3, 4, 0 and the sum of their remainders is 1 + 2 + 3 + 4 + 0 = 10.

46, 47, 48, 49, 50 have remainders 1, 2, 3, 4, 0 and the sum of their remainders is 1 + 2 + 3 + 4 + 0 = 10.

10 * 10 = 100

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Re: What is the sum of the remainders when the first 50 positive integers [#permalink]

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12 Nov 2017, 23:27
(0+1+2+3+4) ten times
equal to 100
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Re: What is the sum of the remainders when the first 50 positive integers   [#permalink] 12 Nov 2017, 23:27
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