Other Topics Puzzle – Challenge 99

Challenge:

Find the highest four-digit number that is divisible by each of the numbers 16, 36, 45, and 80.

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For a positive integer to be divisible by those numbers, it must be divisible by their Least common multiple:
Factorize the numbers to find their Least Common Multiple.
\(16 = 2^4\)
\(36 = 2^2 × 3^2\)
\(45 = 3^2 × 5\)
\(80 = 2^4 × 5\)
Least Common Multiple \(= 2^4 × 3^2 × 5 = 720\)
Every multiple of 720 is divisible by all those above.
Every multiple of 720 can be written as 720n, where n is an integer. The largest 4-digit integer is 9999. Therefore:
\(720n ≤ 9999 → n ≤ \frac{9999}{720} → n ≤ 13.8875 → n = 13 →
720n = 9360\)

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