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# The sum of all integers from 43 to 107, inclusive, is divisible by whi

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Joined: 02 Sep 2009
Posts: 52431
The sum of all integers from 43 to 107, inclusive, is divisible by whi  [#permalink]

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26 Nov 2017, 08:26
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Difficulty:

45% (medium)

Question Stats:

64% (01:54) correct 36% (02:47) wrong based on 58 sessions

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The sum of all integers from 43 to 107, inclusive, is divisible by which of the following numbers?

A. 17
B. 18
C. 21
D. 24
E. 39

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The sum of all integers from 43 to 107, inclusive, is divisible by whi  [#permalink]

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26 Nov 2017, 08:44
1
Bunuel wrote:
The sum of all integers from 43 to 107, inclusive, is divisible by which of the following numbers?

A. 17
B. 18
C. 21
D. 24
E. 39

Total numbers from 43 to 107 $$= 107-43+1=65$$

Sum of nos from 43 to 107 $$= \frac{65}{2}(43+107) = 65*75 = 3*5^3*13$$

from among the options only $$39$$ divides it.

Option E
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The sum of all integers from 43 to 107, inclusive, is divisible by whi  [#permalink]

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26 Nov 2017, 09:57
Bunuel wrote:
The sum of all integers from 43 to 107, inclusive, is divisible by which of the following numbers?

A. 17
B. 18
C. 21
D. 24
E. 39

Sum of n positive integers is $$\frac{n(n+1)}{2}$$

The sum of all integers between 43 and 107 can be found out as follows:

Sum of all positive integers till 107 is $$107*54$$
Similarly, the sum of all positive integers till 42 is $$43*21$$
Therefore, the sum of all integers from 43 to 107 is $$107*54 - 43*21 = 5778 - 903 = 4875$$

$$4875$$ when prime factorized is $$3*5^3*13$$ and the only number which divides this is 39(Option E)
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The sum of all integers from 43 to 107, inclusive, is divisible by whi &nbs [#permalink] 26 Nov 2017, 09:57
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