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# If S and T are positive integers, what are the unique prime factors of

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If S and T are positive integers, what are the unique prime factors of  [#permalink]

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Updated on: 01 Dec 2017, 09:28
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If S and T are positive integers, what are the unique prime factors of product of S and T ?

(1) LCM of S and T is 120
(2) GCF of S and T is 6

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Originally posted by adkikani on 01 Dec 2017, 09:24.
Last edited by Bunuel on 01 Dec 2017, 09:28, edited 1 time in total.
Renamed the topic, edited the question and edited the tags.
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If S and T are positive integers, what are the unique prime factors of  [#permalink]

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01 Dec 2017, 09:40
If S and T are positive integers, what are the unique prime factors of product of S and T ?

(1) LCM of S and T is 120
(2) GCF of S and T is 6

Let's assume that $$S = p_1^a*p_2^b$$ where $$p_1$$ & $$p_2$$ are some prime nos

and $$T=p_1*p_3$$, where $$p_1$$ & $$p_3$$ are prime nos

so LCM of $$S$$ & $$T = p_1^a*p_2^b*p_3$$

and product of $$S$$ & $$T = p_1^{a+1}*p_2^b*p_3$$

so we see that LCM has all the prime factors that the product of the number has. Hence if we know the LCM we can calculate the prime factors of the product

Statement 1: Sufficient

Statement 2: Insufficient

Option A
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Re: If S and T are positive integers, what are the unique prime factors of  [#permalink]

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01 Dec 2017, 10:58
If S and T are positive integers, what are the unique prime factors of product of S and T ?

(1) LCM of S and T is 120
(2) GCF of S and T is 6

We can solve this using numbers..

1. Lets take set of numbers with LCM 120 - (30,40) (24,10) (60,120).. The Unique Prime Numbers will be always 2,3,5
2. Lets take another set with GCF as 6 (12,18) - Prime Factors will be 2,3.
Another set with GCF as 6 (24,30) - Prime Factors will be 2,3,5.. So Insufficient.
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Re: If S and T are positive integers, what are the unique prime factors of  [#permalink]

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01 Dec 2017, 16:53
niks18
Bunuel
VeritasPrepKarishma

Quote:
Let's assume that $$S = p_1^a*p_2^b$$ where $$p_1$$ & $$p_2$$ are some prime nos

and $$T=p_1*p_3$$, where $$p_1$$ & $$p_3$$ are prime nos

Can you explain why is there a difference in notation for S and T, meaning how did
you assume $$T=p_1*p_3$$
For all I know, S and T can very well be two distinct prime numbers.
Should not T be expressed as $$T=p_3^x*p_4^y$$

The basic definition of LCM and GCF comes as follows:
a. For LCM, we break down a no in to its prime factors and list highest powers
of ALL prime factors.
b. For HCF, we list for least power of common prime factors.

I picked up completely opposite answer here ie B instead of A with my above understanding.
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If S and T are positive integers, what are the unique prime factors of  [#permalink]

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01 Dec 2017, 18:30
If S and T are positive integers, what are the unique prime factors of product of S and T ?

(1) LCM of S and T is 120
(2) GCF of S and T is 6

Hi..

If you look at the GCF for the answer, you miss out on prime factors that are not common to both..
Product of two number=LCM*GCF
And LCM already has GCF in it, that is LCM=GCF*x*y ..., Where x ,y etc are prime numbers of just one of them.
So knowing just the LCM is sufficient
Statement I gives the LCM
Suff
A
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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

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If S and T are positive integers, what are the unique prime factors of  [#permalink]

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01 Dec 2017, 19:40
chetan2u wrote:
If S and T are positive integers, what are the unique prime factors of product of S and T ?

(1) LCM of S and T is 120
(2) GCF of S and T is 6

Hi..

If you look at the FACT for the answer, you miss out on prime factors that are not common to both..
Product of two number=LCM*GCF
And LCM already has GCF in it, that is LCM=GCF*x*y ..., Where x ,y etc are prime numbers of just one of them.
So knowing just the LCM is sufficient
Statement I gives the LCM
Suff
A

Hi chetan2u

Can you please let me know if there is anything wrong with my approach above. I thought we can do it by just substituting some numbers.

Sent from my Lenovo P1a42 using GMAT Club Forum mobile app
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If S and T are positive integers, what are the unique prime factors of  [#permalink]

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01 Dec 2017, 20:04
niks18
Bunuel
VeritasPrepKarishma

Quote:
Let's assume that $$S = p_1^a*p_2^b$$ where $$p_1$$ & $$p_2$$ are some prime nos

and $$T=p_1*p_3$$, where $$p_1$$ & $$p_3$$ are prime nos

Can you explain why is there a difference in notation for S and T, meaning how did
you assume $$T=p_1*p_3$$
For all I know, S and T can very well be two distinct prime numbers.
Should not T be expressed as $$T=p_3^x*p_4^y$$

The basic definition of LCM and GCF comes as follows:
a. For LCM, we break down a no in to its prime factors and list highest powers
of ALL prime factors.
b. For HCF, we list for least power of common prime factors.

I picked up completely opposite answer here ie B instead of A with my above understanding.

you can assume any prime factor for either of the numbers but as you have mentioned the process for calculating LCM & HCF, use it to find LCM, HCF & the product.

So let's take your example for $$T$$

$$S = p_1^a*p_2^b$$

$$T=p_3^x*p_4^y$$

LCM of $$S$$ & $$T=p_1^a*p_2^b*p_3^x*p_4^y$$

HCF of $$S$$ & $$T=1$$ as no prime factors are common

Product of $$S$$ & $$T=p_1^a*p_2^b*p_3^x*p_4^y$$

Now you can see that all prime factors in the LCM are there in the product of these two numbers.

you can take any other example for S & T or may be assume simple numbers for S & T to arrive at the answer.

Point is the definition of LCM clearly states that it is the LEAST COMMON MULTIPLE of all the prime factors of the given number and when you multiple those given numbers it will automatically contain the prime factors of the numbers.
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Re: If S and T are positive integers, what are the unique prime factors of  [#permalink]

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01 Dec 2017, 21:40
rahul16singh28 wrote:
chetan2u wrote:
If S and T are positive integers, what are the unique prime factors of product of S and T ?

(1) LCM of S and T is 120
(2) GCF of S and T is 6

Hi..

If you look at the FACT for the answer, you miss out on prime factors that are not common to both..
Product of two number=LCM*GCF
And LCM already has GCF in it, that is LCM=GCF*x*y ..., Where x ,y etc are prime numbers of just one of them.
So knowing just the LCM is sufficient
Statement I gives the LCM
Suff
A

Hi chetan2u

Can you please let me know if there is anything wrong with my approach above. I thought we can do it by just substituting some numbers.

Sent from my Lenovo P1a42 using GMAT Club Forum mobile app

Hi...
The method is correct but would be lengthy and susceptible to errors as it will require calculations 3-4 times.
But is apt if you don't know the exact formula.
Try to go by method if you know them as it will give you confidence and surety.
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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

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Re: If S and T are positive integers, what are the unique prime factors of &nbs [#permalink] 01 Dec 2017, 21:40
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