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# Is the positive integer x prime?

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Math Expert
Joined: 02 Sep 2009
Posts: 56261
Is the positive integer x prime?  [#permalink]

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31 Jul 2018, 22:03
00:00

Difficulty:

45% (medium)

Question Stats:

62% (01:16) correct 38% (00:54) wrong based on 98 sessions

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Is the positive integer x prime?

(1) The Greatest Common Factor of x and y is 1.

(2) The Least Common Multiple of x and y is xy.

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Re: Is the positive integer x prime?  [#permalink]

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31 Jul 2018, 22:11
1
Bunuel wrote:
Is the positive integer x prime?

(1) The Greatest Common Factor of x and y is 1.

(2) The Least Common Multiple of x and y is xy.

Statement 1:
Suppose x=9 ,y=2 GCF = 1 . X is not prime
If x=2, y=9 GCF = 1 . X is prime
Insufficient

Statement2:
Suppose x=9 ,y=2 LCM = xy = 18 . X is not prime
If x=2, y=9 LCM = xy = 18 . X is prime
Insufficient

1&2: Give same data

Hence E
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Re: Is the positive integer x prime?  [#permalink]

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01 Aug 2018, 00:41
Hi,

Again this question, if we know the below simple properties then it’s just a 20-sec problem.

Product of two numbers = Products of their HCF and LCM

• If two numbers are m and n, then HCF of m and n cannot be larger than the difference between m and n

• Two consecutive numbers (x and x+1) can share no factors other than 1, HCF will be 1 and LCM will be x(x + 1).

Two numbers are called co-prime(relatively prime) if HCF of two numbers is 1 and LCM is product of two numbers.

For two numbers to be co-prime not necessary that numbers has to be prime.

If you don’t know the below above rule, then don’t hesitate plug in numbers.

Question is “x” prime ?

That’s is, whether x = 2,3,5,7,11, ….

Statement I is insufficient:

The Greatest Common Factor of x and y is 1.

If x = 2 and y =3 , then G.C.F is 1 and answer to the question is YES(x is prime).

But if x = 4 and y = 3, then G.C.F is 1 and answer to the question is NO(x is not prime).

Statement II is insufficient:

The Least Common Multiple of x and y is xy.

We can take the same examples,

If x = 2 and y =3 , then L.C.M is 6 and answer to the question is YES(x is prime).

But if x = 4 and y = 3, then L.C.M is 12 and answer to the question is NO(x is not prime).

Together also we can use the same numbers,

So not sufficient.

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Re: Is the positive integer x prime?  [#permalink]

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18 Aug 2018, 21:30
E

We can use the property of Co-primes .
Q: is X prime?
Does X have only 2 factors Viz 1 and itself?
Does X^2 have only 3 factors?

St1. NOT SUFFICIENT
X and Y are consecutive integers-
Property- Consecutive integers have only 1 as a common factor. Hence a pair of consecutive integers is co-prime.
X=2 Y=3 >> X is prime
x=4 y=5 >> X is not a prime

st.2 - NOT SUFFICIENT
LCM OF X anf y =xy

Same property of co primes- use x any y as consecutive integers. You'll get 2 cases. So insufficient

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Re: Is the positive integer x prime?   [#permalink] 18 Aug 2018, 21:30
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