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30, 40 and 'n' are such that every number is a factor of the product

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Bunuel
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30, 40 and 'n' are such that every number is a factor of the product [#permalink] New post   02 Dec 2021, 10:00
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30, 40 and 'n' are such that every number is a factor of the product of the other two numbers. If 'n' is a positive integer, what is the difference between the maximum and minimum value of 'n' ?

A. 0
B. 1
C. 134
D. 1134
E. 1188
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E
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Anurocks1233
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Re: 30, 40 and 'n' are such that every number is a factor of the product [#permalink] New post   02 Dec 2021, 10:17
Minimum n is 12
Max 1200
Hence 1200-12 = 1188

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Sushen
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Re: 30, 40 and 'n' are such that every number is a factor of the product [#permalink] New post   02 Dec 2021, 18:00
Rewrite the problem in a pedantic way so that it is easily understood:
30 is a factor of 40n
40 is a factor of 30n
n is a factor of 1200
1200
n’s greatest value is easy now… the greatest factor of 1200 is 1200

Now, let’s find the smallest value:
Prime factorization makes this task relatively easy.
30 = 2*3*5 40 = 2^3 * 5

30 is a factor of 40n: look at what factors are in 30 that are missing in 40. Those factors must be in n! -----> n needs a factor of 3 in it.
40 is a factor of 30n: Do the same. n needs a factor of 2^2 in it.

The smallest n is 2^2 * 3 = 12

1200 - 12 = 1188

E is the answer!
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Re: 30, 40 and 'n' are such that every number is a factor of the product [#permalink] New post   04 Dec 2021, 13:39
Bunuel wrote:
30, 40 and 'n' are such that every number is a factor of the product of the other two numbers. If 'n' is a positive integer, what is the difference between the maximum and minimum value of 'n' ?

A. 0
B. 1
C. 134
D. 1134
E. 1188


Isn't 1 the smallest positive factor of every number? I believe the answer should have been 1200 - 1 = 1199
IanStewart, your thoughts? Am I missing anything here?
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Re: 30, 40 and 'n' are such that every number is a factor of the product [#permalink] New post   04 Dec 2021, 16:40
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errorlogger wrote:
Isn't 1 the smallest positive factor of every number?


Yes, that's certainly true. The question isn't properly worded, so it might be confusing, but we're looking for more than that here. The question means to ask for the smallest value of n which will make it true that each of the three numbers n, 30 and 40 is a factor of the product of the other two numbers. So we need it to be true that n is a factor of 30*40 (and then n could be 1) but we also need it to be true that 30 is a factor of 40n (and now n cannot be 1) and also that 40 is a factor of 30n (and again, n can't be 1). We also need to find the largest possible such value of n, and find the difference, but Sushen's post above yours is perfect and detailed, so I'll suggest you read that if you need a complete solution.
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Re: 30, 40 and 'n' are such that every number is a factor of the product [#permalink] New post   30 Apr 2024, 01:56
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Re: 30, 40 and 'n' are such that every number is a factor of the product [#permalink]
30 Apr 2024, 01:56
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