For any even integer P, 300 multiplied by P is square of an integer. What is the least value of P?
A) 3
B) 4
C) 10
D) 12
E) 14
OA: D
5-Day Free Trial
5-day free, full-access trial TTP
Available with Beat the GMAT members only code
MORE DETAILS
Question on Perfect Square 8
This topic has expert replies
-
richachampion
- Legendary Member
- Posts: 698
- Joined: Tue Jul 21, 2015 12:12 am
- Location: Noida, India
- Thanked: 32 times
- Followed by:26 members
- GMAT Score:740
R I C H A,
My GMAT Journey: 470 → 720 → 740
Target Score: 760+
[email protected]
1. Press thanks if you like my solution.
2. Contact me if you are not improving. (No Free Lunch!)
My GMAT Journey: 470 → 720 → 740
Target Score: 760+
[email protected]
1. Press thanks if you like my solution.
2. Contact me if you are not improving. (No Free Lunch!)
-
richachampion
- Legendary Member
- Posts: 698
- Joined: Tue Jul 21, 2015 12:12 am
- Location: Noida, India
- Thanked: 32 times
- Followed by:26 members
- GMAT Score:740
Not a tough question, but question stem is slightly Tricky.
R I C H A,
My GMAT Journey: 470 → 720 → 740
Target Score: 760+
[email protected]
1. Press thanks if you like my solution.
2. Contact me if you are not improving. (No Free Lunch!)
My GMAT Journey: 470 → 720 → 740
Target Score: 760+
[email protected]
1. Press thanks if you like my solution.
2. Contact me if you are not improving. (No Free Lunch!)
-
fiza gupta
- Master | Next Rank: 500 Posts
- Posts: 216
- Joined: Sun Jul 31, 2016 9:55 pm
- Location: Punjab
- Thanked: 31 times
- Followed by:7 members
300*p(even integer) = perfect square
let y = 100*3*P
100 is a perfect square but not 3
to make y a perfect square P should be 3 or a multiple of 3
p=3 or p =3*z(z a perfect square)(3*4,3*9.....)
but p is even so
p = 3*4(12),3*16(48)...........
only option D satisfies
let y = 100*3*P
100 is a perfect square but not 3
to make y a perfect square P should be 3 or a multiple of 3
p=3 or p =3*z(z a perfect square)(3*4,3*9.....)
but p is even so
p = 3*4(12),3*16(48)...........
only option D satisfies
Fiza Gupta
GMAT/MBA Expert
-
[email protected]
- Elite Legendary Member
- Posts: 10392
- Joined: Sun Jun 23, 2013 6:38 pm
- Location: Palo Alto, CA
- Thanked: 2867 times
- Followed by:511 members
- GMAT Score:800
Hi richachampion,
This question is based on the same concept as the other prime-factor questions that you posted: when you prime-factor a perfect square, each of the prime factors MUST show up an EVEN number of times...
eg. 9 = (3)(3) here, there are TWO 3s.
eg. 100 = (2)(2)(5)(5) here, there are TWO 2s and TWO 5s
eg. 16 = (2)(2)(2)(2) = here, there are FOUR 2s
Etc.
We're told that P is an EVEN INTEGER and that (300)(P) is a PERFECT SQUARE. We have to be careful to incorporate ALL of the information that we're given.. Prime-factoring this number will get us...
(300)(P) =
(10)(30)(P) =
(2)(5)(2)(3)(5)(P) =
(2)(2)(3)(5)(5)(P)
We have TWO 2s and TWO 5s, but only ONE 3. We need there to be an even number of each prime factor. At first glance, you might think that P must equal 3... However, 3 is NOT an EVEN INTEGER. So we need P to contain a '3' AND and at least TWO 2s (so that P will be EVEN).
Final Answer: D
GMAT assassins aren't born, they're made,
Rich
This question is based on the same concept as the other prime-factor questions that you posted: when you prime-factor a perfect square, each of the prime factors MUST show up an EVEN number of times...
eg. 9 = (3)(3) here, there are TWO 3s.
eg. 100 = (2)(2)(5)(5) here, there are TWO 2s and TWO 5s
eg. 16 = (2)(2)(2)(2) = here, there are FOUR 2s
Etc.
We're told that P is an EVEN INTEGER and that (300)(P) is a PERFECT SQUARE. We have to be careful to incorporate ALL of the information that we're given.. Prime-factoring this number will get us...
(300)(P) =
(10)(30)(P) =
(2)(5)(2)(3)(5)(P) =
(2)(2)(3)(5)(5)(P)
We have TWO 2s and TWO 5s, but only ONE 3. We need there to be an even number of each prime factor. At first glance, you might think that P must equal 3... However, 3 is NOT an EVEN INTEGER. So we need P to contain a '3' AND and at least TWO 2s (so that P will be EVEN).
Final Answer: D
GMAT assassins aren't born, they're made,
Rich
-
Matt@VeritasPrep
- GMAT Instructor
- Posts: 2630
- Joined: Wed Sep 12, 2012 3:32 pm
- Location: East Bay all the way
- Thanked: 625 times
- Followed by:119 members
- GMAT Score:780
The OA (and the explanations above) are not quite correct.
Let's set p = 2k, where k is some unknown integer.
300 * p = a square
300 * 2k = a square
600k = a square
(100) * 6k = a square
Since 100 is a square, 6k must also be a square. If k = 0, then 6k is a square, and we're set. Any k < 0 won't work, since squares are nonnegative.
Since p = 2k, p = 0 is our smallest solution, and we're done!
(Unfortunately, whoever wrote this question forgot to put the correct answer. :/ But if we want p > 0, you can see that the next 6k that is a square is 6*6, which gives k = 6 and p = 2*k = 2*6 = 12.)
Let's set p = 2k, where k is some unknown integer.
300 * p = a square
300 * 2k = a square
600k = a square
(100) * 6k = a square
Since 100 is a square, 6k must also be a square. If k = 0, then 6k is a square, and we're set. Any k < 0 won't work, since squares are nonnegative.
Since p = 2k, p = 0 is our smallest solution, and we're done!
(Unfortunately, whoever wrote this question forgot to put the correct answer. :/ But if we want p > 0, you can see that the next 6k that is a square is 6*6, which gives k = 6 and p = 2*k = 2*6 = 12.)
-
Matt@VeritasPrep
- GMAT Instructor
- Posts: 2630
- Joined: Wed Sep 12, 2012 3:32 pm
- Location: East Bay all the way
- Thanked: 625 times
- Followed by:119 members
- GMAT Score:780
Beware!richachampion wrote:Not a tough question, but question stem is slightly Tricky.
The question was subtle enough to fool whoever wrote it (p = 0 is an annoying detail, but exactly the sort of annoying detail that the GMAT likes to test!) ... on the GMAT, if it seems easy, be very suspicious!
GMAT/MBA Expert
-
Scott@TargetTestPrep
- GMAT Instructor
- Posts: 7243
- Joined: Sat Apr 25, 2015 10:56 am
- Location: Los Angeles, CA
- Thanked: 43 times
- Followed by:29 members
If 300P is a perfect square, then its prime factorization must contain only even exponents. Let's begin by prime factoring 300.richachampion wrote:For any even integer P, 300 multiplied by P is square of an integer. What is the least value of P?
A) 3
B) 4
C) 10
D) 12
E) 14
OA: D
300 = 30 x 10 = 5 x 6 x 5 x 2 = 5 x 3 x 2 x 5 x 2 = 5^2 x 3^1 x 2^2
We can see that 300 is not a perfect square because its prime factorization contains an odd exponent (that is, 3^1). This means that P must contribute the missing factor. However, we are given that P must also be an even integer. Thus, in order to keep "an even number" of twos, P must be divisible by 2^2. Thus P must be divisible by 3^1 x 2^2 = 12. Since we want the least value of P, P must be 12.
In fact, if P = 12, then 300P = 5^2 x 3^2 x 2^4, which is a perfect square.
Answer: D
Scott Woodbury-Stewart
Founder and CEO
[email protected]
See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews
