A basketball club has between 31 and 39 members. When they line up in pairs nobody is left over, but when they line up in groups of 3 exactly one person is left over. How many members are there? Choose from 32, 34, 35 or 36.

Step-by-step solution

1. 1

   Translate each sentence into a divisibility statement. Let $n$ be the number of members. Then

   $31\le n\le 39,\qquad n\equiv 0\pmod 2,\qquad n\equiv 1\pmod 3.$

2. 2

   Apply the first condition to shrink the list. The even numbers in range are

   $32,\ 34,\ 36,\ 38.$

   This already eliminates option C ($35$), which is odd.

3. 3

   Apply the second condition. Divide each survivor by $3$ and record the remainder:

   $32=3(10)+2,\quad 34=3(11)+1,\quad 36=3(12)+0,\quad 38=3(12)+2.$

   Only $34$ leaves remainder $1$.

4. 4

   Check the answer against both original sentences. $34$ members form $17$ pairs with nobody spare ✓, and $34$ members form $11$ groups of three with one person left over ✓. So the club has $34$ members — option B.

5. 5

   Note the general structure. The two conditions together say $n\equiv 4\pmod 6$, since $n$ must be even and $\equiv 1\pmod 3$. The numbers satisfying that are $\ldots,28,34,40,\ldots$, and exactly one of them, $34$, lies in the given range — which is why the problem has a unique answer.