How many of the first 500 positive integers are divisible by 3, 4 and 5?

Question

Mathematics

Celeste Compton

12 May, 18:03

How many of the first 500 positive integers are divisible by 3, 4 and 5?

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Answers (2)

Know the Answer?

Madelynn Bishop · Answer 1

The smallest positive integer is:

3 * 4 * 5 = 60

Therefore: 60, 120, 180, 240, 300, 360, 420, 480

Jake Dunlap · Answer 2

In order for an integer to be divisible by 3 4 and 5, it has to be divisible by the least common multiple of the 3, which is 3*4*5 = 60. So, the problem is now "how many of the first 500 positive integers are divisible by 60?"

You can answer this question by setting up an inequality: in order for a number to be positive and a multiple of 60, it has to be equal to 60 * (an integer). So, if n is a positive integer, you have

60n ≤ 500 ... now solve for n by division

n ≤ 8.333

Now the question is "how many positive integers are less than (or equal to) 8.333?" Clearly, all the numbers 1-8 are less than 8.333 while 9 is too high. Therefore, you have 8 positive integers, which means there are 8 multiples of 60, which means there are 8 positive integers divisible by 3, 4 and 5 that are less than or equal to 500.