Question

Find the number of positive integers not exceeding 10,000 that are not divisible by 3, 7, or 8.

Answer 1

The number of positive integers not exceeding 10,000 that are not divisible by 3, 7, or 8 is 5000. This can be obtained by using the inclusion-exclusion property.

Find the number of positive integers:

Here in the question it is given that,

• Numbers should be between 1 to 10,000  

• Numbers should be not divisible by 3, 7, or 8

We have to find the number of positive integers not exceeding 10,000 that are not divisible by 3, 7, or 8.

For that we first find the number of positive integers not exceeding 10,000 that are divisible by 3, 7, or 8.

By inclusion-exclusion property we can write that,

$[\frac{10000}{3} ]+[\frac{10000}{7} ]+[\frac{10000}{8} ]-[\frac{10000}{21} ]-[\frac{10000}{24} ]-[\frac{10000}{56} ]+[\frac{10000}{168} ]$

= 3333 + 1428 + 1250 - 476 - 416 - 178 - 59

= 6011 - 1070 - 59

= 5000

Therefore there are 5000 numbers which are divisible by 3, 7 and 8.

So, the number of positive integers not exceeding 10,000 that are not divisible by 3, 7, or 8 will be,

10,000 - 5000 = 5000

Hence the number of positive integers not exceeding 10,000 that are not divisible by 3, 7, or 8 is 5000.

Learn more about inclusion-exclusion property here:

brainly.com/question/10927267

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