2006 AMC 10B Problem 22

Problem 22 of 25HarderNumber Theory

Elmo makes

N

sandwiches for a fundraiser. For each sandwich he uses

B

globs of peanut butter at

4

¢ per glob and

J

blobs of jam at

5

¢ per blob. The cost of the peanut butter and jam to make all the sandwiches is

\$2.53.

Assume that

B,

J,

and

N

are positive integers with

N \gt 1.

What is the cost of the jam Elmo uses to make the sandwiches?

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Solution

The total cost is

N(4B+5J)=253

cents

=11\cdot23.

Since

N \gt 1,

N\in\{11,23,253\}.

N=253

N=23,

then

4B+5J

equals

1

11,

impossible for positive integers. So

N=11

and

4B+5J=23,

whose only positive solution is

B=2,

J=3.

The jam costs

11\cdot3\cdot5=165

cents, or

\$1.65.

Thus, the correct answer is D.