2520 is the smallest number that can be divided by each of t

2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.

What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?

Solution

Write each number from 2 to 20 as products of primes:
2 = 2¹
3 = 3¹
4 = 2²
5 = 5¹
6 = 2¹ x 3¹
7 = 7¹
8 = 2³
9 = 3²
10 = 2¹ x 5¹
11 = 11¹
12 = 2² x 3¹
13 = 13¹
14 = 2¹ x 7¹
15 = 3¹ x 5¹
16 = 2
17 = 17¹
18 = 2¹ x 3²
19 = 19¹
20 = 2² x 5¹

Now take the highest power of each prime factor, and multiply together

2 x 3² x 5¹ x 7¹ x 11¹ x 13¹ x 17¹ x 19¹ = 232,792,560

the smallest number divisible by 1, 2, 3, ... , 20 is 232792560