The least number, divisible by all the natural numbers from 1 to 10, is

Concept: The least common multiple (LCM) of several positive integers is the smallest positive integer divisible by every one of them.

To obtain it, write each number in prime-power form and take each prime at the greatest exponent needed by any number.

Application: For the numbers 1 through 10, determine the highest required power of each prime and multiply those powers.

1. The required highest prime powers are 23 (from 8), 32 (from 9), 5, and 7.

2. LCM(1, 2, …, 10) = 23 × 32 × 5 × 7 = 8 × 9 × 5 × 7.

3. Multiplying in order gives 8 × 9 = 72, 72 × 5 = 360, and 360 × 7 = 2520.

Cross-check: Direct divisibility checks confirm the selected prime powers:

• 2520 ÷ 8 = 315.

• 2520 ÷ 9 = 280.

• 2520 ÷ 5 = 504.

• 2520 ÷ 7 = 360.

Every integer from 1 through 10 uses no prime power greater than those selected, so each divides 2520; removing any required prime factor or power would break divisibility.

Result: The least number divisible by every natural number from 1 through 10 is 2520.