Let \( x \) be the number of clients interviewed each day. On the first day, clients are interviewed in groups of 4, which implies that \( x \) must be divisible by 4:
\[ x \equiv 0 \mod 4 \]
On the second day, clients are interviewed in groups of 8, meaning \( x \) must also be divisible by 8:
\[ x \equiv 0 \mod 8 \]
To satisfy both conditions, \( x \) must be a common multiple of 4 and 8. The least common multiple (LCM) of 4 and 8 can be calculated as follows:
First, we find the prime factorizations:
- \( 4 = 2^2 \)
- \( 8 = 2^3 \)
The LCM is determined by taking the highest power of each prime factor present in the factorizations. Here, the only prime is \( 2 \):
\[ \text{LCM}(4, 8) = 2^3 = 8 \]
Thus, the smallest number of clients that could be interviewed each day is:
\[ \boxed{8} \]
To confirm, we check that 8 is divisible by both 4 and 8:
- \( 8 \div 4 = 2 \) (which is an integer)
- \( 8 \div 8 = 1 \) (which is also an integer)
Therefore, the solution is valid, and the minimum number of clients interviewed each day is indeed \( \boxed{8} \).
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