Problem of the Day


John and Deborah work together at their college library. If John works on a given day, he won’t work again until the j^{th} day after that. If Deborah works on a given day, she won’t work again until the d^{th} day after that. In dj-7 days, the semester will end and John and Deborah will no longer work at the library. If they both work ¬†today, will they both work on the same day again before the semester ends?

(1) j is even, and d is odd

(2)  None of the prime factors that divide j evenly divide d evenly.

Reveal Answer

Answer

B. Statement 2 alone is sufficient, but statement 1 alone is not sufficient.

See the Solution

Solution

[latexpage]

If John works the $j^{th}$ day after every day that he works, and he works today, then he will work on the day that is $j$, $2j$, $3j$… days from today. That is, the number of days until some future work day is always a ¬†multiple of $j$ if he works today. Similarly, Deborah’s work days will be some multiple of $d$ days from today. Thus, if they both work today, they won’t work together again until some number of days has passed that is a multiple of BOTH $j$ and $d$. The first such day will be the least common multiple of $j$ and $d$. If $j$ and $d$ have no factors in common, then their least common multiple will be $jd$. Otherwise, it will be smaller. For example, if $j=3$ and $d=8$, LCM$(3,8)=3 \cdot 8=24$, because 3 and 8 do not share any factors. However, if $j=2$ and $d=4$, then LCM$(2,4)=4$, which is not that same as $2 \cdot 4$.

Statement 1: $j$ is even, and $d$ is odd.¬†If $j=3$ and $d=8$, then the semester will end in 17 days, but they won’t work together for another 24 days. However, if $j=3$ and $d=6$, then the semester will end in 11 days, but they will work together in 6 days. INSUFFICIENT.

Statement 2: ¬†¬†¬†None of the prime factors that divide $j$ evenly divide $d$ evenly. This tells us that $j$ and $d$ have no common factors, which means LCM$(j,d)=jd$. The semester ends in $jd-7$ days, but they won’t work together again for $jd$ more days. Thus, they definitely won’t work together again. SUFFICIENT.


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