Finding Numbers Divisible by 6 and 8

To find the numbers between 1 and 100 that are divisible by both 6 and 8, we need to find numbers divisible by their Least Common Multiple (LCM).

Calculating the LCM of 6 and 8

First, find the prime factorization of each number:

• $6 = 2 \times 3$
• $8 = 2^3$

The LCM is found by taking the highest power of each prime factor present:

LCM(6, 8) = $2^3 \times 3 = 8 \times 3 = 24$

Identifying Multiples of 24

We need to count the multiples of 24 that fall within the range of 1 to 100.

List the multiples of 24:

• $24 \times 1 = 24$
• $24 \times 2 = 48$
• $24 \times 3 = 72$
• $24 \times 4 = 96$

The next multiple, $24 \times 5 = 120$, is greater than 100 and thus outside the specified range.

Counting the Numbers

The numbers between 1 and 100 divisible by both 6 and 8 are 24, 48, 72, and 96.

There are 4 such numbers.