How Many Numbers Between 20 and 100 Are Divisible by Both 3 and 7?

To determine how many numbers between 20 and 100 are divisible by both 3 and 7, we first need to understand a key concept: the least common multiple (LCM). The LCM of 3 and 7 is a number that is divisible by both, and it will help us identify all the qualifying numbers within a given range.

Calculating the Least Common Multiple (LCM)

The LCM of 3 and 7 can be found by multiplying the two numbers together, since they are both prime and have no common factors other than 1:

LCM(3, 7) 3 x 7 21.
This means that any number divisible by both 3 and 7 will also be divisible by 21.

Identifying Multiples of 21

Next, we need to find the multiples of 21 that lie within the range from 20 to 100. We start by identifying the smallest multiple of 21 that is greater than or equal to 20:

21 x 1 21

Then, we identify the largest multiple of 21 that is less than or equal to 100:

Divide 100 by 21: 100 / 21 4 with a remainder Therefore, the largest multiple is 21 x 4 84

Now, we list the multiples of 21 within this range:

21 x 1 21 21 x 2 42 21 x 3 63 21 x 4 84

Counting these multiples, we find there are 4 numbers between 20 and 100 that are divisible by both 3 and 7. These numbers are 21, 42, 63, and 84.

Summary

Therefore, there are 4 numbers between 20 and 100 that are divisible by both 3 and 7. To find this, we determined the LCM of 3 and 7, which is 21, and then calculated the multiples of 21 within the given range.

Numbers that are divisible by both 3 and 7 will always be divisible by their least common multiple, 21. Finding the multiples of 21 within a specific range involves simple arithmetic and the understanding of the LCM concept.

Additional Insight

There are a total of 80 integer numbers between 20 and 100. Only 4 of these numbers are divisible by both 3 and 7, as we determined through the process above.

The formula to verify this calculation can be expressed as:

floor(100/21) - floor(20/21) floor(4.76) - floor(0.95) 4 - 0 4

Thus, there are indeed 4 numbers between 20 and 100 that are divisible by both 3 and 7.