How many numbers from 1 to 100 are multiples of 2 3 4 6 8 9?

Answer 1

Assuming you mean that you want the number of multiples of each, then for 1-100:

number of multiples of 2 = 50

number of multiples of 3 = 33

number of multiples of 4 = 25

number of multiples of 6 = 16

number of multiples of 8 = 12

number of multiples of 9 = 11

Assuming you mean that you want the numbers that are multiples of 2, 3, 4, 6, 8 or 9, then some numbers may be multiples of more than one (for example 12 is a multiple of 2, 3, 4 and 6) and so a straight addition of the number of multiples of each cannot be done:

Consider 2, 4 and 8

Every multiple of 4 or 8 is also a multiple of 2, so all the multiples of 4 and 8 are counted by the multiples of 2.

Consider 3 and 9

Every multiple of 9 is also a multiple of 3, so all the multiples of 9 are counted by the multiple of 3

Consider 2, 3 and 6.

Every multiple of 6 is an even multiple of 3, so are counted in both the multiples of 2 and 3.

So the total number of multiples of 2, 3, 4, 6, 8 or 9 is the number of multiples of 2 plus the number of multiples of 3 minus the number of multiples of 6:

For 1 to 100,

Number of multiples of 2 = 50

Number of multiples of 3 = 33

Number of multiples of 6 = 16

So number of multiples of 2, 3, 4, 6, 8 or 9 in 1-100 is 50+33-16 = 67.

Assuming you mean that they are multiples of all of 2, 3, 4, 6, 8 and 9, then they must be multiples of the lowest common multiple of 2, 3, 4, 6 ,8, 9

2 = 21, 3 = 31, 4 = 22, 6 = 2131, 8 = 23, 9 = 32

LCM = highest power of the primes used = 2332 = 72

Thus all numbers that are multiples of 2, 3, 4, 6, 8 and 9 are multiples of 72, which means between 1 and 100 only 1 number is a multiple of all of them, namely 72