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Which of the following numbers is a perfect square?

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emmak
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Which of the following numbers is a perfect square? [#permalink] New post   Updated on: 22 Mar 2013, 03:21
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Which of the following numbers is a perfect square?

A. 1266
B. 1444
C. 2022
D. 4034
E. 8122
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B

Originally posted by emmak on 22 Mar 2013, 01:04.
Last edited by Bunuel on 22 Mar 2013, 03:21, edited 1 time in total.
Renamed the topic and edited the question.
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Re: Which of the following numbers is a perfect square? [#permalink] New post   22 Mar 2013, 03:40
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emmak wrote:
Which of the following numbers is a perfect square?

A. 1266
B. 1444
C. 2022
D. 4034
E. 8122


A perfect square, is an integer that is the square of an integer. For example 16=4^2, is a perfect square.

From above we can deduce that the units digit of a perfect square cannot be 2, 3, or 7. Discard C and E.

Another property: perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: \(36=2^2*3^2\), powers of prime factors 2 and 3 are even.

Make prime factorization of the options:

A. 1266 = 2*3*211. Discard. We could discard 1266 after we got that 1266 = 2*633 = 2*odd, so 2 in 1266 has an odd power, which means that 1266 is NOT a prefect square.

B. 1444 = 2^2*19^2 --> so, 1444 IS a perfect square.

D. 4034 = 2*2017. Discard. We could discard 4034 after we got that 4034 = 2*2017 = 2*odd, so 2 in 4034 has an odd power, which means that 4034 is NOT a prefect square.

Answer: B.

Hope it's clear.
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Re: Which of the following numbers is a perfect square? [#permalink] New post   05 Feb 2014, 14:27
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emmak wrote:
Which of the following numbers is a perfect square?

A. 1266
B. 1444
C. 2022
D. 4034
E. 8122


Let's begin by eliminating answer choices C and E since a perfect square cannot end in 2,3,7 or 8

Now between A,B and D

Well we can see that all numbers are even for starters. This means that they also must be divisible by 4

Only B fits the bill

Hence B is the correct answer

Hope its clear
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J
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Re: Which of the following numbers is a perfect square? 1266 [#permalink] New post   22 Mar 2013, 01:20
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perfect Square never ends with 2, 3, 7 and 8 so straight away you can eliminate 8122 and 2022.
38*38 = 1444
hence B.
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Re: Which of the following numbers is a perfect square? [#permalink] New post   04 Mar 2014, 02:45
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Perfect squares cannot end with 2,3,7 or 8, so options C & E are eliminated

A. 1266
B. 1444
E. 8122

Look at the above 3 options. All are divisible by 2, but only 1444 is divisible by 4.
Hence Answer = B

NB - Consider last two digits of any number. If Last two digits of any number are divisible by 4, then the complete number is divisible by 4
Also, If Last digit of any number is divisible by 2, then the complete number is divisible by 3
If Last three digits of any number are divisible by 8, then the complete number is divisible by 8
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Re: Which of the following numbers is a perfect square? [#permalink] New post   25 Jun 2014, 01:48
Well we can see that all numbers are even for starters. This means that they also must be divisible by 4

Only B fits the bill

Hence B is the correct answer

Dear J,

Can you please explain the underlined portion ? It'd be really helpful
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Re: Which of the following numbers is a perfect square? [#permalink] New post   25 Jun 2014, 02:47
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Tanvr wrote:
Well we can see that all numbers are even for starters. This means that they also must be divisible by 4

Only B fits the bill

Hence B is the correct answer

Dear J,

Can you please explain the underlined portion ? It'd be really helpful


For an even number to be a prefect square it must be a multiple of 4. That's because we know that a prefect square must have an even powers of its primes, so 2 in even number must have even power to be a prefect square: 2, 4, 6, ... so in any case it must be a multiple of 4.
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Which of the following numbers is a perfect square? [#permalink] New post   Updated on: 17 Oct 2022, 00:51
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Tanvr wrote:
Well we can see that all numbers are even for starters. This means that they also must be divisible by 4

Only B fits the bill

Hence B is the correct answer

Dear J,

Can you please explain the underlined portion ? It'd be really helpful


Perfect squares have even powers of prime factors. What this means is that if a number is a perfect square, and it has 2 as a prime factor, the power of 2 in the number will be even i.e. it will have two 2s or four 2s or six 2s etc. Similarly, if it has 3 as a factor, it will have two 3s or four 3s or six 3s etc.

Now when you see that 2 is a factor of all leftover options, you know that you will have at least two 2s i.e. the number must be divisible by 4 if the number is to be a perfect square. Check the last two digits of the numbers. If the last two digits are divisible by 4, the number will be divisible by 4. Only option (B) is divisible by 4 (because 44 is divisible by 4). Hence (B) is the correct answer.

Originally posted by KarishmaB on 25 Jun 2014, 02:49.
Last edited by KarishmaB on 17 Oct 2022, 00:51, edited 1 time in total.
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Re: Which of the following numbers is a perfect square? [#permalink] New post   23 Jul 2014, 08:25
Quick solution: B

the perfect square of whatever a integer number has the following possible digit 1 ; 4 ; 5 ; 6; 9
=> eliminate: C and E
A and D is divisible by 2, but not by 4 => A and D is not a perfect square

B is correct
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Re: Which of the following numbers is a perfect square? [#permalink] New post   25 Jun 2019, 23:40
All the numbers in the options are even numbers.
For an even number to be even, it should be divisible by 4. We can check divisibility by 4 by only looking at the last 2 digits of the number.
(A number id divisible by 4 if it's last 2 digits are a multiple of 4).

Option B. 1444 -> last 2 digits make 44 -> 44 is divisible by 4. Hence option B is divisible by 4 and is a possible candidate for a perfect square. None of the other even numbers are divisible by 4, so they CANNOT be perfect squares.

Answer:Option B
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Re: Which of the following numbers is a perfect square? [#permalink] New post   09 Sep 2019, 10:59
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emmak wrote:
Which of the following numbers is a perfect square?

A. 1266
B. 1444
C. 2022
D. 4034
E. 8122


The units digit of a perfect square can be only 0, 1, 4, 5, 6, or 9. Therefore, we can eliminate choices C and E. Since 30^2 = 900 and 40^2 = 1,600, if 1266 and 1444 are perfect squares, they must be the square of a number between 30 and 40. Since 1266 is closer to 1600 than it is to 900 and if it’s a perfect square, only one integer - 36 - could be its square root..However, since 36^2 = 1296, then 1266 is not a perfect square. Similarly, since 1444 is closer to 1600 than it is to 900 and if it’s a perfect square, only one integer - 38 - could be its square root. Since 38^2 = 1444, we see that 1444 is a perfect square.

Answer: B
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Re: Which of the following numbers is a perfect square? [#permalink] New post   05 Dec 2020, 07:32
Perfect squares need to have pairs of prime factors. If we divide a number by 2 and it's odd, it cannot be a perfect square. Lets take a look at the answer choices:

A. 1266 /2 = 633
B. 1444 / 2 = 722
C. 2022 / 2 = 1011
D. 4034 / 2 = 2017
E. 8122 / 2 = 4061

Once B is even; therefore B must be the answer.
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Re: Which of the following numbers is a perfect square? [#permalink] New post   15 Jan 2024, 04:17
Bunuel wrote:
emmak wrote:
Which of the following numbers is a perfect square?

A. 1266
B. 1444
C. 2022
D. 4034
E. 8122


A perfect square, is an integer that is the square of an integer. For example 16=4^2, is a perfect square.

From above we can deduce that the units digit of a perfect square cannot be 2, 3, or 7. Discard C and E.

Another property: perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: \(36=2^2*3^2\), powers of prime factors 2 and 3 are even.

Make prime factorization of the options:

A. 1266 = 2*3*211. Discard. We could discard 1266 after we got that 1266 = 2*633 = 2*odd, so 2 in 1266 has an odd power, which means that 1266 is NOT a prefect square.

B. 1444 = 2^2*19^2 --> so, 1444 IS a perfect square.

D. 4034 = 2*2017. Discard. We could discard 4034 after we got that 4034 = 2*2017 = 2*odd, so 2 in 4034 has an odd power, which means that 4034 is NOT a prefect square.

Answer: B.

Hope it's clear.



By the same token, would it be fair to say that “if the number of distinct factors of a perfect square is ODD, it is a perfect square”?
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Re: Which of the following numbers is a perfect square? [#permalink] New post   15 Jan 2024, 07:45
emmak wrote:
Which of the following numbers is a perfect square?

A. 1266
B. 1444
C. 2022
D. 4034
E. 8122

Square numbers end with 0, 1, 4, 5, 6 or 9 at the unit's place. (Reject options C & E)

Now, use square rule of 2 Digit no having unts digit as 5...

35*35 = 1225
45*45 = 2025
55*55 = 3035
65*65 = 4225

Now, check the optins....

(A) 1266 , units digit must be 4 for having square and units digit as 6

\(35*35 = 1225 < 36*36 = 1296\)

Thus, 1266 is not possible...

(B) 1444 , units digit must have 2 or 8

35*35 = 1225 < 38*38 = 1444, Answer....

Once can check the other options, Answer must be (B)
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Re: Which of the following numbers is a perfect square? [#permalink] New post   15 Jan 2024, 14:33
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Ophelia__ wrote:
Bunuel wrote:
emmak wrote:
Which of the following numbers is a perfect square?

A. 1266
B. 1444
C. 2022
D. 4034
E. 8122


A perfect square, is an integer that is the square of an integer. For example 16=4^2, is a perfect square.

From above we can deduce that the units digit of a perfect square cannot be 2, 3, or 7. Discard C and E.

Another property: perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: \(36=2^2*3^2\), powers of prime factors 2 and 3 are even.

Make prime factorization of the options:

A. 1266 = 2*3*211. Discard. We could discard 1266 after we got that 1266 = 2*633 = 2*odd, so 2 in 1266 has an odd power, which means that 1266 is NOT a prefect square.

B. 1444 = 2^2*19^2 --> so, 1444 IS a perfect square.

D. 4034 = 2*2017. Discard. We could discard 4034 after we got that 4034 = 2*2017 = 2*odd, so 2 in 4034 has an odd power, which means that 4034 is NOT a prefect square.

Answer: B.

Hope it's clear.



By the same token, would it be fair to say that “if the number of distinct factors of a perfect square is ODD, it is a perfect square”?


Yes, every number with an odd number of positive factors is indeed a perfect square.
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