Which of The Following Numbers is A Perfect Square? GMAT Problem Solving

Question: Which of the following numbers is a perfect square?

1. 1266
2. 1444
3. 2022
4. 4034
5. 8122

Correct Answer: B
Approach Solution (1):
A perfect square, is an integer that is square of an integer. For example\(16 =4^2\),is a perfect square
Form above, we can deduce that the unit’s digit of a perfect square cannot be 2, 3, or 7. Discard C and E.
Another property: perfect square always has even power of its prime factors. The reverse is also true: if a number has even powers of its prime factor 2 and 3 are even.
Make prime factorization of the options:

1. 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 perfect square.
2. 1444 = \(2^2*19^2\rightarrow\) so, 1444 is a perfect square.
3. 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 perfect square.

Approach Solution (2):
Look at the above 3 options. All are divisible by 2, but only 1444 is divisible by 4.
Hence answer is 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 3.
If last three digits of any number are divisible by 8, the the complete number is divisible by 8.

“Which of the following numbers is a perfect square?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills.

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