Question

Compute how many integers between 1 to 1000 are no divisible by 2,3,5,or 7

Answer 1

Hi there!
Here's the answer:

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SOLUTION :

Here there are two cases:

¶¶¶ Case - 1:
Both the No.s in the Given Range are Inclusive, which are divisible by 2, 3, 5, & by 7 respectively.

We want

| (A∪B∪C ∪D)' |

Now,

| (A∪B∪C ∪D)' | = | A | + | B | + | C | + | D | - | A∩B | - | B∩C | - | C∩D | + | A∩B∩C | + | B∩C∩D | + | C∩D∩A | - | A∩B∩C∩D |

$| A | = [\frac{1000}{2}]$ = 500

$| B | = [\frac{1000}{3}]$ = 333

$| C | = [\frac{1000}{5}]$ = 200

$| D | = [\frac{1000}{7}]$ = 142

$| A∩B | = [\frac{1000}{6}]$ = 166

$| A∩C | = [\frac{1000}{10}]$ = 100

$| A∩D | = [\frac{1000}{14}]$ = 71

$| B∩C | = [\frac{1000}{15}]$ = 66

$| B∩D | = [\frac{1000}{21}]$ = 47

$| C∩D | = [\frac{1000}{35}]$ = 28



$| A∩B∩C | = [\frac{1000}{30}]$ = 33

$| A∩B∩D | = [\frac{1000}{42}]$= 23

$| A∩C∩D | = [\frac{1000}{70}]$ = 14

$| B∩C∩D | = [\frac{1000}{100}]$ = 9



$| A∩B∩C∩D | = [\frac{1000}{210}]$ = 4


So

| (A∪B∪C ∪D) | = 772

But The required answer is 1000 - 772 = 228
(As we require its complement value)


¶¶¶ Case - 2:

When both the No.s in range are excluded

1 is not divisible by 2, 3, 5 & 7 while 1000 is divisible by 5

•°• The answer is 228 - 1 = 227


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Hope it helps