1995 AJHSME Problems/Problem 12

Problem

A lucky year is one in which at least one date, when written in the form month/day/year, has the following property: The product of the month times the day equals the last two digits of the year. For example, 1956 is a lucky year because it has the date 7/8/56 and $7\times 8 = 56$ . Which of the following is NOT a lucky year?

$\text{(A)}\ 1990 \qquad \text{(B)}\ 1991 \qquad \text{(C)}\ 1992 \qquad \text{(D)}\ 1993 \qquad \text{(E)}\ 1994$

Solution

We examine only the factors of $90, 91, 92, 93,$ and $94$ that are less than $13$ , because for a year to be lucky, it must have at least one factor between $1$ and $12$ to represent the month.

$90$ has factors of $1, 2, 3, 5, 6, 9,$ and $10$ . Dividing $90$ by the last number $10$ , gives $9$ . Thus, $10/9/90$ is a valid date, and $1990$ is a lucky year.

$91$ has factors of $1$ and $7$ . Dividing $91$ by $7$ gives $13$ , and $7/13/91$ is a valid date. Thus, $1991$ is a lucky year.

$92$ has factors of $1, 2$ and $4$ . Dividing $92$ by $4$ gives $23$ , and $4/23/92$ is a valid date. Thus, $1992$ is a lucky year.

$93$ has factors of $1$ and $3$ . Dividing $93$ by $3$ gives $31$ , and $3/31/93$ is a valid date. Thus, $1993$ is a lucky year.

$94$ has factors of $1$ and $2$ . Dividing $94$ by $2$ gives $47$ , and no month has $47$ days. Thus, $1994$ is not a lucky year, and the answer is $\boxed{E}$ .

1995 AJHSME (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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1995 AJHSME Problems/Problem 12

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