Smallest Integer Divisible by All Numbers from 1 to 100

Theorem

The smallest positive integer which is divisible by each of the integers from $1$ to $100$ is:

$69 \, 720 \, 375 \, 229 \, 712 \, 477 \, 164 \, 533 \, 808 \, 935 \, 312 \, 303 \, 556 \, 800$

Proof

Let $N$ be divisible by each of the integers from $1$ to $100$.

Each prime number between $2$ and $97$ must be a divisor of $N$.

Also:

$2^6 = 64 \divides N$

$3^4 = 81 \divides N$

$5^2 = 25 \divides N$

$7^2 = 49 \divides N$

Every other integer between $1$ and $100$ is the product of a subset of all of these.

Hence by Euclid's Lemma:

\(\ds N\)

\(\ge\)

\(\ds 2^6 \times 3^4 \times 5^2 \times 7^2 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29 \times 31 \times 37 \times 41 \times 43 \times 47 \times 53 \times 59 \times 61 \times 67 \times 71 \times 73 \times 79 \times 83 \times 89 \times 97\)

\(\ds \)

\(=\)

\(\ds 69 \, 720 \, 375 \, 229 \, 712 \, 477 \, 164 \, 533 \, 808 \, 935 \, 312 \, 303 \, 556 \, 800\)

$\blacksquare$

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $69,720,375,229,712,477,164,533,808,935,312,303,556,800$