What is the least common multiple of the numbers from 1 to 10?

Answer 1

We need to find the least common multiple of 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. We can find the least common multiple of smaller sets of numbers, and then compare those least common multiples.

Any number is a multiple of 1, so we do not need to consider it further.

The least common multiple of 2, 4, and 8 is 8, since 8 is a multiple of both 2 and 4.

The least common multiple of 3 and 9 is 9, since 9 is a multiple of 3.

The least common multiple of 5 and 10 is 10, since 10 is a multiple of 5.

So, now we have reduced the set of numbers to 6, 7, 8, 9, and 10.

To determine the least common multiple of more than two numbers, we can take them two at a time, then compare those least common multiples to find their least common multiple. To determine the least common multiple of two numbers, determine the prime factors of both numbers. Then, determine the prime factors they have in common. Multiply all their prime factors together (in other words, multiply both numbers together) and divide by the prime factors they have in common (in other words, their greatest common factor).

The least common multiple of 6 and 7 is 42, since their greatest common factor is 1.

The least common multiple of 8 and 9 is 72, since their greatest common factor is 1.

So, now we have reduced the set of numbers to 10, 42, and 72.

The greatest common factor of 10 and 42 is 2, so the least common multiple is 10 x 42 ÷ 2 = 210.

Now, we only need to determine the least common multiple of 72 and 210.

First, determine the greatest common factor of both numbers.

The prime factors of 72 are 2, 2, 2, 3, and 3.

The prime factors of 210 are 2, 3, 5, and 7.

The prime factors they have in common are 2 and 3, so the greatest common factor is 2 x 3 = 6.

The least common multiple of 72 and 210 is 72 x 210 ÷ 6 = 2520.

Doublecheck:

2520 ÷ 10 = 252 = 22 x 32 x 7

2520 ÷ 9 = 280 = 22 x 5 x 7

2520 ÷ 8 = 315 = 32 x 5 x 7

2520 ÷ 7 = 360 = 23 x 32 x 5

2520 ÷ 6 = 420

2520 ÷ 5 = 504

2520 ÷ 4 = 630

2520 ÷ 3 = 840

2520 ÷ 2 = 1260

2520 ÷ 1 = 2520

From the prime factorization of the first four division results above, you can see that they do not have a prime factor common to all, which confirms that we have the least common multiple.

LCM for 10, 9, 8, 7, 6, 5, 4, 3, 2, and 1 is 2520, which is the multiple of the highest power of prime factors in the given numbers (23 x 32 x 5 x 7).