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

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Is the sum of the integers from 54 to 153 inclusive, divisib [#permalink] New post 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.
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Re: Is the sum of the integers from 54 to 153 inclusive.... [#permalink] New post 20 Sep 2013, 16:09
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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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Re: Is the sum of the integers from 54 to 153 inclusive.... [#permalink] New post 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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Re: Is the sum of the integers from 54 to 153 inclusive.... [#permalink] New post 20 Sep 2013, 17:20
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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.
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Re: Is the sum of the integers from 54 to 153 inclusive.... [#permalink] New post 21 Sep 2013, 02:36
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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, divisib [#permalink] New post 01 Sep 2015, 23:14
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Re: Is the sum of the integers from 54 to 153 inclusive, divisib   [#permalink] 01 Sep 2015, 23:14
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