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11 Jun 2017, 12:00
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In one of the MGMAT books:
"If n is the product of 2,3, and a two-digit prime number, how many of its factors are greater than 6?"
In the answer they say:
"Because we have been asked for a concrete answer, we can infer that the answer will be the same
regardless of which 2-digit prime we pick."
Does anyone know why the answer doesn't depend on which 2-digit prime is picked?
Thanks.
"If n is the product of 2,3, and a two-digit prime number, how many of its factors are greater than 6?"
In the answer they say:
"Because we have been asked for a concrete answer, we can infer that the answer will be the same
regardless of which 2-digit prime we pick."
Does anyone know why the answer doesn't depend on which 2-digit prime is picked?
Thanks.
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11 Jun 2017, 21:25
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omegan3
It all boils down to prime factorization. A prime number has no other factors so if you break down the number into prime factors, you will find they will always be 2, 3 and that prime number. Hence, in the product, all factors of the product of 2 and 3 i.e. 6 will have 1 corresponding factor in the entire product. No more, no less.
2 x 3 x 11 = 66 -> Factors: 1, 2, 3, 6, 11, 22, 33, 66
2 x 3 x 23 = 138 -> Factors: 1, 2, 3, 6, 23, 46, 69, 138
2 and 3 multiply to become 6 and its essentially 6 x any two digit prime number which doesn't have any factors other than 1 that it shares with 6 and itself. The number of factors greater than 6 will be corresponding to all factors between 1 and 6 i.e. 4.
12 Jun 2017, 03:57
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The question says the number has ONLY three factors 2, 3 and 2-digit prime number.
The factors of the number will then be 1, 2, 3 and the 2-digit prime number. So, what ever 2-digit prime number you pick (11,13,17, 19 etc.) you will only have that 2-digit prime number greater than 6.
Now, had the question been modified such that instead of 2-digits, it was a single digit prime number, then the answer will be different. If that prime number was 5, in that case we will have been zero factors above 6. But if the prime number was 7, then would have one factor above 6.
Hope this helps!
The factors of the number will then be 1, 2, 3 and the 2-digit prime number. So, what ever 2-digit prime number you pick (11,13,17, 19 etc.) you will only have that 2-digit prime number greater than 6.
Now, had the question been modified such that instead of 2-digits, it was a single digit prime number, then the answer will be different. If that prime number was 5, in that case we will have been zero factors above 6. But if the prime number was 7, then would have one factor above 6.
Hope this helps!
