6 digit divisibility

Number Theory Level pending

Which is the greatest 6-digit number divisible by 4, 8, 10, 6, 9, 15, 45, 24, 36 and 20?


The answer is 999720.

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1 solution

Chew-Seong Cheong
Apr 11, 2020

When a number N N is divisible by 4 , 6 , 8 , 9 , 10 , 15 , 20 , 24 , 36 , 4, 6, 8, 9, 10, 15, 20, 24, 36, and 45 45 means that N N is divisible by the lowest common multiple of 4 , 6 , 8 , 9 , 10 , 15 , 20 , 24 , 36 , 4, 6, 8, 9, 10, 15, 20, 24, 36, and 45 45 . And lcm ( 4 , 6 , 8 , 9 , 10 , 15 , 20 , 24 , 36 , 45 ) = 2 3 × 3 2 × 5 = 360 \text{lcm }(4, 6, 8, 9, 10, 15, 20, 24, 36, 45) = 2^3 \times 3^2 \times 5 = 360 .

The greatest 6-digit divisible by 360 360 is 1000000 360 × 360 = 2777 × 360 = 999720 \left \lfloor \dfrac {1000000}{360} \right \rfloor \times 360 = 2777 \times 360 = \boxed{999720} .

Notation: \lfloor \cdot \rfloor denotes the floor function .

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