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# Counting problem

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How many three-digit numbers are multiples of neither 5 nor 8?

Jul 24, 2022

#1
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The smallest multiple of 5 is $$100 \Rightarrow (20 \times 5)$$, and the largest multiple of 5 is $$995 \Rightarrow (199 \times 5)$$. There are $$199 - 20 + 1 = 180$$ multiples of 5

The smallest multiple of 8 is $$104 \Rightarrow (13 \times 8)$$, and the largest multiple of 5 is $$995 \Rightarrow (124 \times 8)$$. There are $$124 - 13 + 1 = 122$$ multiples of 8

However, we overcount every $$\text{lcm}(5,8) = 40$$ numbers.

The smallest number we overcount for is $$120 \Rightarrow (3 \times 40)$$, and the largest number is $$960 \Rightarrow (24 \times 40)$$. There are $$24 - 3 + 1 = 22$$ numbers we overcount for.

Can you take it from here?

Jul 24, 2022
#3
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There are: 900 / 5 ==180 multiples of 5

There are: 900 / 8 ==112 multiples of 8

There are: 900 / 40 ==22 multiples of 40

900 - 180 - 112 + 22 ==630 integers that are neither multiples of 5 or 8

Jul 24, 2022