Discussion

What is the greatest five-digit number that is completely divisible by 8, 15, 16 and 21?

Answer: B

A number divisible by 8, 15, 16 and 21 must be divisible by their least common multiple.

\[ 8 = 2^3,\ 15 = 3 \times 5,\ 16 = 2^4,\ 21 = 3 \times 7 \]

\[ \text{LCM} = 2^4 \times 3 \times 5 \times 7 = 1680 \]

The greatest five digit number is 99999. Dividing, \(99999 = 1680 \times 59 + 879\), so the largest multiple not above it is

\[ 1680 \times 59 = 99120 \]

The number 987640 has six digits, 95780 is not a multiple of 1680 and 79685 is not even a multiple of 8. The answer is 99120, option B.

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