SOLUTION: A student wants to know how many integers between 1 and 1000 are a multiple of 4 or a multiple of 5. She wonders if it is correct to find the number of those integers that are mul

Question 278810: A student wants to know how many integers between 1 and 1000 are a multiple of 4 or a multiple of 5. She wonders if it is correct to find the number of those integers that are multiples of 4 and add the number of those that are multiples of 5. How do you respond?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
not quite.

there are certain numbers that are both multiples of 5 and multiples of 4 and those need to be subtracted from the total because they will be counted twice.

for example:

assume the numbers were from 1 to 100 (we can count those easier.

100 / 4 = 25
100 / 5 = 20
100 / 20 = 5

The total of numbers should be 25 + 20 - 5 = 40 numbers.

we would have:

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

20 is counted twice so we have to subtract 1 of them.
40 is counted twice so we have to subtract 1 of them.
60, 80, and 100 are counted twice so we have to subtract 1 of them.

the same principle applies from 1 to 1000.

number of 4's is 1000/4 = 250
number of 5's is 1000/5 = 200
number of 20's (4*5) is 1000/20 = 50
total is 250 + 200 - 50 = 400

bottom line is you have to subtract the numbers that are both divisible by 4 and 5 to avoid them from being double counted.

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