It's True By Induction

Number Theory Level 1

The smallest positive integer that is divisible by 3, 4 and 5 is 3 × 4 × 5 = 60 3\times4\times5=60 .

Is it also true that the smallest positive integer that is divisible by 3, 4, 5 and 6 is 3 × 4 × 5 × 6 = 360 3\times4\times5\times6=360 ?

Yes, it is true No, it is not true

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Pi Han Goh by Pi Han Goh , 30, Malaysia

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2 solutions

Relevant wiki: Lowest Common Multiple

lcm ( 3 , 4 , 5 , 6 ) = 3 × lcm ( 1 , 4 , 5 , 2 ) = 3 × 2 × lcm ( 1 , 2 , 5 , 1 ) = 3 × 2 × 2 × 5 = 60 \text{lcm}(3,4,5,6)=3 \times \text{lcm}(1,4,5,2)=3 \times 2 \times\text{lcm}(1,2,5,1)= 3 \times 2 \times 2 \times 5=60 .

In general LCM = product of numbers if and only if they are pairwise coprime.

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It would be easier to notice that (it is already given) 3 × 4 × 5 = 60 3\times4\times5=60 is already divisible by 6, so the smallest positive number that divisible by these 4 numbers is still 60.

Pi Han Goh - 5 years ago

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It would be a bit tedious to do this way if there were many numbers.

A Former Brilliant Member - 5 years ago

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Yes, I didn't that my method works for all kinds of questions.

Pi Han Goh - 5 years ago
Ratnesh Singh
Jun 9, 2016

3 *4 *5 *6=factorise=3 *(2 *2) *5 *(3 *2)=make pairs and write once=(3 *3) *(2 *2) *2 *5=3 *2 *2 *5=60

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