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# Is the sum of the integers from 54 to 153 inclusive, divisib

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Joined: 06 Sep 2013
Posts: 2044
Concentration: Finance
GMAT 1: 770 Q0 V
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Kudos [?]: 302 [0], given: 354

Is the sum of the integers from 54 to 153 inclusive, divisib [#permalink]  20 Sep 2013, 16:01
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Is the sum of the integers from 54 to 153 inclusive, divisible by 100?

Hint: Can be solved fast with a property.
Math Expert
Joined: 02 Sep 2009
Posts: 27494
Followers: 4312

Kudos [?]: 42304 [0], given: 6012

Re: Is the sum of the integers from 54 to 153 inclusive.... [#permalink]  20 Sep 2013, 16:09
Expert's post
jlgdr wrote:
Is the sum of the integers from 54 to 153 inclusive, divisible by 100?

Hint: Can be solved fast with a property.

# of integers from 54 to 153 inclusive is 153-54+1=100.

The sum = (average)*(# of integers) = (54+153)/2*100=103.5*100=10350 --> not a multiple of 100.
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Kudos [?]: 1732 [0], given: 123

Re: Is the sum of the integers from 54 to 153 inclusive.... [#permalink]  20 Sep 2013, 16:15
jlgdr wrote:
Is the sum of the integers from 54 to 153 inclusive, divisible by 100?

Hint: Can be solved fast with a property.

My other approach

Sum of 1 to 153, inclusive = [(1+153)/2] x 153 = 77 x 153
Sum of 1 to 53, inclusive = [(1+53)/2] x 53 = 27 x 153

Sum of 54 to 153, inclusive = 77*153 - 27*53
= 77*100 + 77*53 - 27*53
= 77*100 + 53*(77-27)
= 77*100 + 53*50

Only 77*100 is divisible by 100

==> The ans is: NOT divisible by 100

Hope it helps.
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SVP
Joined: 06 Sep 2013
Posts: 2044
Concentration: Finance
GMAT 1: 770 Q0 V
Followers: 26

Kudos [?]: 302 [1] , given: 354

Re: Is the sum of the integers from 54 to 153 inclusive.... [#permalink]  20 Sep 2013, 17:20
1
KUDOS
Also by Property:

For any set of consecutive integers with an EVEN number of items, the sum of all the items is NEVER a multiple of the number of items.
Math Expert
Joined: 02 Sep 2009
Posts: 27494
Followers: 4312

Kudos [?]: 42304 [0], given: 6012

Re: Is the sum of the integers from 54 to 153 inclusive.... [#permalink]  21 Sep 2013, 02:36
Expert's post
jlgdr wrote:
Also by Property:

For any set of consecutive integers with an EVEN number of items, the sum of all the items is NEVER a multiple of the number of items.

Correct.

Properties of consecutive integers:
• If n is odd, the sum of n consecutive integers is always divisible by n. Given $$\{9,10,11\}$$, we have $$n=3=odd$$ consecutive integers. The sum is 9+10+11=30, which is divisible by 3.
• If n is even, the sum of n consecutive integers is never divisible by n. Given $$\{9,10,11,12\}$$, we have $$n=4=even$$ consecutive integers. The sum is 9+10+11+12=42, which is NOT divisible by 4.

For more check here: math-number-theory-88376.html
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Re: Is the sum of the integers from 54 to 153 inclusive....   [#permalink] 21 Sep 2013, 02:36
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# Is the sum of the integers from 54 to 153 inclusive, divisib

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