A performance group has between 24 and 44 members. Arranged in groups of 5 nobody is left over, and arranged in groups of 7 there are 2 people left over. How many members are there?

Step-by-step solution

1. 1

   Translate both conditions into congruences. 'Rows of $5$ with nobody left over' means the count $n$ satisfies $n\equiv0\pmod 5$. 'Rows of $7$ with $2$ left over' means $n\equiv2\pmod 7$. Together with $24\le n\le44$, these pin down $n$ completely — two congruences with coprime moduli have one solution per $35$ consecutive integers, and the range spans $21$.

2. 2

   List the candidates from the stricter, easier condition. Multiples of $5$ between $24$ and $44$:

   $25,\ 30,\ 35,\ 40$

   Starting from the divisibility condition rather than the remainder condition keeps the list short — only four numbers to test instead of six.

3. 3

   Test each against division by $7$.

   $25=7\cdot3+4,\qquad 30=7\cdot4+2\ \checkmark,\qquad 35=7\cdot5+0,\qquad 40=7\cdot5+5$

   Only $30$ leaves a remainder of $2$.

4. 4

   Confirm uniqueness. By the Chinese Remainder Theorem the pair $n\equiv0\pmod5$, $n\equiv2\pmod7$ has a single solution modulo $35$, namely $n\equiv30\pmod{35}$. The next candidates are $30-35=-5$ and $30+35=65$, both outside $[24,44]$, so $30$ is the only answer.

5. 5

   Check against the original wording. $30\div5=6$ rows exactly, with no one left over ✓. $30\div7=4$ rows of seven with $30-28=2$ people left over ✓. Both conditions hold, and $30$ lies inside the stated range, so the answer is choice A.