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