# 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 $\boxed{E}$.