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# X is a number which on squaring produces Y. If Y has 3 factors, how

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Joined: 04 Jan 2015
Posts: 2313
X is a number which on squaring produces Y. If Y has 3 factors, how  [#permalink]

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Updated on: 13 Aug 2018, 01:20
00:00

Difficulty:

55% (hard)

Question Stats:

54% (01:35) correct 46% (01:42) wrong based on 135 sessions

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e-GMAT Question:

X is a number which on squaring produces Y. If Y has 3 factors, how many such X are present in the first 20 natural numbers?

A) 2
B) 4
C) 5
D) 7
E) 8

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Question 6 of The e-GMAT Number Properties Marathon

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Question 7 of the Marathon

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Originally posted by EgmatQuantExpert on 27 Feb 2018, 09:40.
Last edited by EgmatQuantExpert on 13 Aug 2018, 01:20, edited 3 times in total.
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Re: X is a number which on squaring produces Y. If Y has 3 factors, how  [#permalink]

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27 Feb 2018, 10:13
1
EgmatQuantExpert wrote:

Question:

X is a number which on squaring produces Y. If Y has 3 factors, how many such X are present in the first 20 natural numbers?

A) 2
B) 4
C) 5
D) 7
E) 8

given $$x^2=y$$

$$y$$ has only 3 factors implies that $$y$$ is a square of a prime number, Hence $$y=p^2$$, where $$p$$ is some prime number

Hence $$x^2=p^2 => x$$ is a prime number.

Prime numbers less than 20 are: 2,3,5,7,11,13,17,19. total $$8$$

Option E
e-GMAT Representative
Joined: 04 Jan 2015
Posts: 2313
Re: X is a number which on squaring produces Y. If Y has 3 factors, how  [#permalink]

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27 Feb 2018, 23:17
1

Solution:

The question specifies that the square of $$X$$: $$Y$$, has $$3$$ factors.
o So, $$1$$ and $$Y$$ must be the factors of $$Y$$.
 Apart from these two, $$Y$$ has one more factor.
o Per the conceptual knowledge of Prime factorization, a number can be represented in the form of: $$P1^a * P2^b * P3^c…Pn^z$$, where $$P1, P2,P3$$.. are the prime factors of the number $$X$$ and $$a, b, c$$.. are the powers of the prime factors.
o Total factors of this number is written as: $$(a+1) * (b+1)$$…
 Since total factors = $$3$$, let us see how can $$3$$ be formed using the above formula:
• $$3 = 3*1$$
This is the only possible combination.
o Now, putting this in the formula of total factors we get:
 $$(a+1) * (b+1) = 3*1$$
 $$(a+1) = 3$$; $$(b+1) =1$$
=> $$a=2, b=0$$
o Thus, the number $$Y$$ can be written as $$P1^2$$ $$* P2^0$$, where $$P1$$and $$P2$$ are the prime factors of the number $$Y$$.
 However, $$P2^0$$ will simplify to $$1$$, which will further reduce the overall equation to: $$Y = P1^2$$.
• Since, we also know that $$Y=X^2$$
• Therefore, $$Y= X^2 = P1^2$$
=>$$X^2 = P1^2$$
=> $$X=P1$$ (Cannot take the negative value since prime numbers are positive numbers)
 This means $$X$$ is nothing but a prime number.
• This was the observation that was needed to solve this question.
• Now, the entire sum reduces to a simple problem of prime numbers, i.e.: How many prime numbers exist between $$1$$ and $$20$$, inclusive?
o Prime numbers between $$1$$ and $$20$$, inclusive, are: $$2,3,5,7,11,13,17,19$$: $$8$$ primes.
Therefore, $$8$$ such numbers lie among the first $$20$$ natural numbers.
_________________

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Guillermo's Success Story | Carrie's Success Story

Ace GMAT quant
Articles and Question to reach Q51 | Question of the week

Number Properties – Even Odd | LCM GCD | Statistics-1 | Statistics-2 | Remainders-1 | Remainders-2
Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2
Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability
Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry
Algebra- Wavy line | Inequalities

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Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets

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Re: X is a number which on squaring produces Y. If Y has 3 factors, how &nbs [#permalink] 27 Feb 2018, 23:17
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