Question

How many numbers are there between 55 and 4505 which are a multiple of 5 and divisible by 3

Answer 1

Answer:

Not sure if the “5 and 3 both” means “both 5 and 3”, or “either 5, 3, or both”, so I’ll show both cases:

For a number to be divisible by 2 numbers, it must be divisible by the products of the 2 numbers. So for a number to be divisible by both 3 and 5, it must be divisible by 15. So we can simply count the number of numbers between 55 and 4495 that are divisible by 15: ⌊4495−max(x:15|x and x<55)15⌋=⌊4495−4515⌋=296

We can use principle of inclusion and exclusion to find the number of numbers between 55 and 4495 that are divisible by 3, 5, or both: Let A be the set of numbers that are divisible by 3 between 55 and 4495, and B be the set of numbers that are divisible by 5 between 55 and 4495. Then by principle of inclusion and exclusion, we get:

|A∪B|=|A|+|B|−|A∩B|=⌊4495−max(x:15|x and x<55)3⌋+⌊4495−max(x:15|x and x<55)5⌋−⌊4495−max(x:15|x and x<55)3⋅5⌋=⌊4495−453⌋+⌊4495−455⌋−⌊4495−4515⌋=1483+890−296=2077

Step-by-step explanation: