1997 AMC 12 Problem 20

Attempt Problem 20 of the 1997 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1997 AMC 12 solutions, or check the answer key.

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20.

Which one of the following integers can be expressed as the sum of

100

consecutive positive integers?

1{,}627{,}384{,}950

2{,}345{,}678{,}910

3{,}579{,}111{,}300

4{,}692{,}581{,}470

5{,}815{,}937{,}260

Write the terms as

a+1,a+2,\ldots,a+100

Their sum must be congruent to

50

modulo

100

The sum is

100a+(1+\cdots+100)

=100a+5050,

so it ends in

50.

Only

1,627,384,950

has that property, and it gives the positive integer

a=\frac{1,627,384,950-5050}{100}.

Thus A is correct.

Problem 20 in other years