# What is th remainder when the sum of the first 102 counting numbers is divided by 5220?

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525181

Created
April 4, 2011 12:11pm UTC

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1. You can find the sum of the first 102 counting numbers using Gauss's method, namely:
"the sum of the first n counting numbers is n*(n+1)/2" whether n is odd or even.

For n = 102, the sum is 102*103/2=10506.
So divide 10506 by 5220 to get a quotient of 2, with a remainder of 66.

Mental calculation tip:
To multiply two numbers close to a hundred can be done in the head as follows, no paper, no calculators:

Let the numbers be (100+x) and (100+y), where x and y are small numbers. Take the example of 102 and 103, then x=2, y=3.
The next two digits are the sum of x and y, that gives 105.
The next two digits are the product of x and y, that gives 10506, et voilà!

Try with 107*109, that should give 11663.

525185

Created
April 4, 2011 12:35pm UTC

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2. Forgot to divide 10506 by 2 according to Gauss's rule:
10506/2=5253, divided by 5220 gives a quotient of 1 with 33 as a remainder, and divided by 5250 gives a quotient of 1 and 3 as a remainder.

525188

Created
April 4, 2011 12:39pm UTC

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3. i have no clue i am tring ti find the answer