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How many positive integers less than 100 are that are NOT multiples of
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Bunuel
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How many positive integers less than 100 are that are NOT multiples of [#permalink] New post  14 Apr 2020, 03:33
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How many positive integers less than 100 are that are NOT multiples of a perfect square greater than 1?

A. 65
B. 62
C. 61
D. 60
E. 52

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chetan2u
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Re: How many positive integers less than 100 are that are NOT multiples of [#permalink] New post  14 Apr 2020, 04:15
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Bunuel wrote:
How many positive integers less than 100 are that are NOT multiples of a perfect square greater than 1?

A. 65
B. 62
C. 61
D. 60
E. 52

Are You Up For the Challenge: 700 Level Questions


We will have to check till \(\sqrt{100}\) or 10.
1) \(2^2=4\).....so \(\frac{100}{4}-1=24\)
2) \(3^2=9\).....so \(\frac{100}{9}=11\), but subtract the common numbers with (1) above.\(.LCM(4,9)=36=>\frac{100}{36}=2\)...Total = 11-2=9
3) \(4^2=16\)...Already included in (1) above
4) \(5^2=25....100/25-1=3\)
5) \(6^2=3^22^2=36\)...Already included in (1)
6) \(7^2=49...\frac{100}{49}=2\)

Total 24+9+3+2=38

Answer 99-38=61

Most important you are looking for <100, so take total as 99
_________________
1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html


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Re: How many positive integers less than 100 are that are NOT multiples of [#permalink] New post  14 Apr 2020, 04:30
Bunuel wrote:
How many positive integers less than 100 are that are NOT multiples of a perfect square greater than 1?

A. 65
B. 62
C. 61
D. 60
E. 52

Are You Up For the Challenge: 700 Level Questions


    Perfect square less than 100 are 4, 9, 16, 25, 36, 49, 64, 81
      16, 36, and 64 are multiple of 4, it means all their multiple will all be multiple of 4.
      36 and 81 is a multiple of 9, all the multiple of 36 and 81 are multiple of 9 as well.
    Now, all the numbers which are multiple of 4, 9, 25, 49 are multiple of perfect square.
      Number of multiple of 4 less than 100 = \(\frac{99}{4} = 24\)
      Number of multiple of 9 less than 100 = \(\frac{99}{9} = 11\)
      Number of multiples of 25 less than 100 = \(\frac{99}{25} = 3\)
      Number of multiple of 49 less than 100 = \(2\)
    Now, one thing 36 is a multiple of 4 as well as 9
      All the multiples of 36 will be multiple of 4 and 9 as well
      We counted the multiple of 36 twice in above calculation
        Multiple of 36 less than 100= \(\frac{99}{36} = 2\)
    Number of integers less than 100, which are multiple of a perfect square = \(24+11+3+2-2=38\)
    Number of integers which are not multiple of perfect square = \(99-38 =61\)
ANSWER: Option C.
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Tiburcio1987
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Re: How many positive integers less than 100 are that are NOT multiples of [#permalink] New post  05 Sep 2020, 00:07
Good solution, quick question, why at the end, with 7^2 you do not do -1, in terms of 100/49=2 and then you do not subtract 1 as you have been doing with the others?



chetan2u wrote:
Bunuel wrote:
How many positive integers less than 100 are that are NOT multiples of a perfect square greater than 1?

A. 65
B. 62
C. 61
D. 60
E. 52

Are You Up For the Challenge: 700 Level Questions


We will have to check till \(\sqrt{100}\) or 10.
1) \(2^2=4\).....so \(\frac{100}{4}-1=24\)
2) \(3^2=9\).....so \(\frac{100}{9}=11\), but subtract the common numbers with (1) above.\(.LCM(4,9)=36=>\frac{100}{36}=2\)...Total = 11-2=9
3) \(4^2=16\)...Already included in (1) above
4) \(5^2=25....100/25-1=3\)
5) \(6^2=3^22^2=36\)...Already included in (1)
6) \(7^2=49...\frac{100}{49}=2\)

Total 24+9+3+2=38

Answer 99-38=61

Most important you are looking for <100, so take total as 99
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chetan2u
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Re: How many positive integers less than 100 are that are NOT multiples of [#permalink] New post  05 Sep 2020, 01:19
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Tiburcio1987 wrote:
Good solution, quick question, why at the end, with 7^2 you do not do -1, in terms of 100/49=2 and then you do not subtract 1 as you have been doing with the others?



chetan2u wrote:
Bunuel wrote:
How many positive integers less than 100 are that are NOT multiples of a perfect square greater than 1?

A. 65
B. 62
C. 61
D. 60
E. 52

Are You Up For the Challenge: 700 Level Questions


We will have to check till \(\sqrt{100}\) or 10.
1) \(2^2=4\).....so \(\frac{100}{4}-1=24\)
2) \(3^2=9\).....so \(\frac{100}{9}=11\), but subtract the common numbers with (1) above.\(.LCM(4,9)=36=>\frac{100}{36}=2\)...Total = 11-2=9
3) \(4^2=16\)...Already included in (1) above
4) \(5^2=25....100/25-1=3\)
5) \(6^2=3^22^2=36\)...Already included in (1)
6) \(7^2=49...\frac{100}{49}=2\)

Total 24+9+3+2=38

Answer 99-38=61

Most important you are looking for <100, so take total as 99


-1 is done only where 100 is a multiple of that number for example 2,4,5, and not with 3 or 7.
Because when we divide 100/2 even 100 is included, but not in case of 100/49 as factors are only 49 and 98.
_________________
1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html


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