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If S1 = {1, 2, 3, 4, ... , 23} and S2 [#permalink]
17 Mar 2016, 07:08
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If S1 = {1, 2, 3, 4, ... , 23} and S2 = {207, 208, 209, 210, 211, ... , 691}, how many elements of the set S2 are divisible by at least four distinct prime numbers that are elements of the set S1? (a) 9 (b) 8 (c) 11 (d) 12 (e) 7
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Re: If S1 = {1, 2, 3, 4, ... , 23} and S2 [#permalink]
17 Mar 2016, 07:14
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Explanation If a number of the set S2 is divisible by at least four distinct prime numbers of the set S1, then it will be divisible by their product as well.  The number of numbers in S2 divisible by the product of 2, 3, 5 and 7, i.e. 210 = 3.
 The number of numbers in S2 divisible by the product of 2, 3, 5 and 11, i.e. 330 = 2.
 The number of numbers in S2 divisible by the product of 2, 3, 5 and 13, i.e. 390 = 1.
 The number of numbers in S2 divisible by the product of 2, 3, 5 and 17, i.e. 510 = 1.
 The number of numbers in S2 divisible by the product of 2, 3, 5 and 19, i.e. 570 = 1.
 The number of numbers in S2 divisible by the product of 2, 3, 5 and 23, i.e. 690 = 1.
 The number of numbers in S2 divisible by the product of 2, 3, 7 and 11, i.e. 462 = 1.
 The number of numbers in S2 divisible by the product of 2, 3, 7 and 13, i.e. 546 = 1.
There is no other combination of four or more prime numbers in set S1 that divides any of the elements of set S2. Hence, the required number of elements = 11.
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Re: If S1 = {1, 2, 3, 4, ... , 23} and S2 [#permalink]
14 Dec 2018, 07:13
soumya1989 wrote: Explanation If a number of the set S2 is divisible by at least four distinct prime numbers of the set S1, then it will be divisible by their product as well.  The number of numbers in S2 divisible by the product of 2, 3, 5 and 7, i.e. 210 = 3.
 The number of numbers in S2 divisible by the product of 2, 3, 5 and 11, i.e. 330 = 2.
 The number of numbers in S2 divisible by the product of 2, 3, 5 and 13, i.e. 390 = 1.
 The number of numbers in S2 divisible by the product of 2, 3, 5 and 17, i.e. 510 = 1.
 The number of numbers in S2 divisible by the product of 2, 3, 5 and 19, i.e. 570 = 1.
 The number of numbers in S2 divisible by the product of 2, 3, 5 and 23, i.e. 690 = 1.
 The number of numbers in S2 divisible by the product of 2, 3, 7 and 11, i.e. 462 = 1.
 The number of numbers in S2 divisible by the product of 2, 3, 7 and 13, i.e. 546 = 1.
There is no other combination of four or more prime numbers in set S1 that divides any of the elements of set S2. Hence, the required number of elements = 11. Why 11? I see you listed 8 elements divisible by 4 or more primes in set S1.



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Re: If S1 = {1, 2, 3, 4, ... , 23} and S2 [#permalink]
17 Dec 2018, 09:58
soumya1989 wrote: Explanation If a number of the set S2 is divisible by at least four distinct prime numbers of the set S1, then it will be divisible by their product as well.  The number of numbers in S2 divisible by the product of 2, 3, 5 and 7, i.e. 210 = 3.
 The number of numbers in S2 divisible by the product of 2, 3, 5 and 11, i.e. 330 = 2.
 The number of numbers in S2 divisible by the product of 2, 3, 5 and 13, i.e. 390 = 1.
 The number of numbers in S2 divisible by the product of 2, 3, 5 and 17, i.e. 510 = 1.
 The number of numbers in S2 divisible by the product of 2, 3, 5 and 19, i.e. 570 = 1.
 The number of numbers in S2 divisible by the product of 2, 3, 5 and 23, i.e. 690 = 1.
 The number of numbers in S2 divisible by the product of 2, 3, 7 and 11, i.e. 462 = 1.
 The number of numbers in S2 divisible by the product of 2, 3, 7 and 13, i.e. 546 = 1.
There is no other combination of four or more prime numbers in set S1 that divides any of the elements of set S2. Hence, the required number of elements = 11. There are so many numbers (more than 400), how do you quickly find the numbers that are divisible by the product of the prime numbers?



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Re: If S1 = {1, 2, 3, 4, ... , 23} and S2 [#permalink]
19 Dec 2018, 21:13
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Think of this we have these prime numbers :2,3,5,7,11,13,17,19,23 the range is from 207 to 691
the product of at least 4 distinct primes sounds scary, but we have the primes all listed out. just multiply the first 4: 2*3*5*7= 210, and there three multiples of 210 within the given bounds, which are: 210,420,630 now try 2*3*5*11=330, two multiples of that are 330 and 660 2*3*5*13= 390 2*3*5*17= 510 2*3*5*19=570 2*3*5*23=690
2*3*7*11=462 2*3*7*13=546
thus we have 11 multiples within the given bounds with at least 4 distinct primes.



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Re: If S1 = {1, 2, 3, 4, ... , 23} and S2 [#permalink]
10 Jan 2019, 10:15
Is there any short way?



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Re: If S1 = {1, 2, 3, 4, ... , 23} and S2 [#permalink]
12 Jan 2019, 10:24
The explanation above is pretty fast. Not all the time there is. Regards
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Re: If S1 = {1, 2, 3, 4, ... , 23} and S2
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12 Jan 2019, 10:24





