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If S1 = {1, 2, 3, 4, ... , 23} and S2

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soumya1989
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If S1 = {1, 2, 3, 4, ... , 23} and S2 [#permalink] New post   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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C
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soumya1989
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Re: If S1 = {1, 2, 3, 4, ... , 23} and S2 [#permalink] New post   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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QuantumWonder
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Re: If S1 = {1, 2, 3, 4, ... , 23} and S2 [#permalink] New post   14 Dec 2018, 07:13
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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] New post   17 Dec 2018, 09:58
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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] New post   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] New post   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] New post   12 Jan 2019, 10:24
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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 [#permalink] New post   06 Jul 2023, 02:22
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