AMC10 2006 B

AMC10 2006 B · Q22

AMC10 2006 B · Q22. It mainly tests Money / coins, Divisibility & factors.

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 > 1. What is the cost of the jam Elmo uses to make the sandwiches?

Elmo为筹款活动做了N个三明治。对于每个三明治，他用了B团花生酱（每团4美分）和J团果酱（每团5美分）。制作所有三明治的花生酱和果酱总成本为2.53美元。假设B、J和N为正整数且N > 1。Elmo制作三明治所用果酱的成本是多少？

(A) $1.05 $1.05

(B) $1.25 $1.25

(D) $1.65 $1.65

(E) $1.85 $1.85

Answer

Correct choice: (D)

正确答案：（D）

Solution

(D) The total cost of the peanut butter and jam is $N(4B+5J)=253$ cents, so $N$ and $4B+5J$ are factors of $253=11\cdot23$. Because $N>1$, the possible values of $N$ are $11$, $23$, and $253$. If $N=253$, then $4B+5J=1$, which is impossible since $B$ and $J$ are positive integers. If $N=23$, then $4B+5J=11$, which also has no solutions in positive integers. Hence $N=11$ and $4B+5J=23$, which has the unique positive integer solution $B=2$ and $J=3$. So the cost of the jam is $11(3)(5\text{¢})=\$1.65$.

（D）花生酱和果酱的总费用为 $N(4B+5J)=253$ 美分，因此 $N$ 和 $4B+5J$ 是 $253=11\cdot23$ 的因数。由于 $N>1$，$N$ 的可能取值为 $11$、$23$ 和 $253$。若 $N=253$，则 $4B+5J=1$，但这不可能，因为 $B$ 和 $J$ 是正整数。若 $N=23$，则 $4B+5J=11$，这在正整数中也无解。因此 $N=11$ 且 $4B+5J=23$，其唯一的正整数解为 $B=2$、$J=3$。所以果酱的价格为 $11(3)(5\text{¢})=\$1.65$。

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