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Re: Which of the following numbers is a perfect square? [#permalink]
22 Mar 2013, 02:40

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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.

Re: Which of the following numbers is a perfect square? [#permalink]
04 Mar 2014, 01:45

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 _________________

Re: Which of the following numbers is a perfect square? [#permalink]
25 Jun 2014, 01:47

Expert's post

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.
_________________

Re: Which of the following numbers is a perfect square? [#permalink]
25 Jun 2014, 01:49

Expert's post

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. The reason for this is explained here: http://www.veritasprep.com/blog/2010/12 ... ly-number/ http://www.veritasprep.com/blog/2010/12 ... t-squares/

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.
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