How many factors of the number 2520 are perfect squares?

Question

A) 5
B) 4
C) 3
D) 2
E) 1

IanStewart · Answer

We can prime factorize 2520:

2520 = 10*252 = (2)(5)(2)(126) = (2)(5)(2)(2)(9)(7) = 2^3 * 3^2 * 5 * 7

Every factor of 2520 thus looks like: 2^a * 3^b * 5^c * 7^d, where a can be 0, 1, 2, or 3, b can be 0, 1 or 2, c can be 0 or 1, and d can be 0 or 1. The only factors that are perfect squares are the ones that have exclusively even exponents. So to make a factor of 2520 that is a perfect square, a can be 0 or 2, for two choices, b can be 0 or 2, for two choices, and c and d must be 0, for one choice each. Multiplying our choices, we have 2*2*1*1 = 4 factors that are perfect squares. If we wanted to list them all, they are (I'll ignore the 5 and 7, since we'll never use them to make a perfect square divisor here) :

2^0 *3^0 = 1
2^2 * 3^0 = 4
2^0 * 3^2 = 9
2^2 * 3^2 = 36

globaldesi · Answer

2520 =  2^3 * 3^2 *5*7 
perfect squares =  2^2 , 1^1 , 3^2 , (2*3) ^2

total = 4
thus B

Kinshook · Answer

Asked: How many factors of the number 2520 are perfect squares?

2520 = 2^3 * 3^2 * 5 * 7

Factors which are perfect squares = (1+2^2)(1+3^2) = {1,4,9,36}

IMO B

Mona2019 · Answer

Last one is (2*3)^2

Posted from my mobile device

Kinshook · Answer

Hi  globaldesi

Minor correction is needed as highlighted in green color.

globaldesi · Answer

Sorry missed the brackets

CrackverbalGMAT · Answer

We can use the rule of counting for factors.

For any number, N broken into prime factors and their powers in the form  N = a^p * b^q*c^r  and so on, then the total number of factors n = (p + 1) * (q + 1) * (r + 1) and so on.

This same rule is taken forward for counting factors which are perfect squares, perfect cubes, and to count factors ending in a zero, 2 zeros and so on. This method becomes useful for large numbers for eg  2^5 * 3^7 * 11^15 , where counting directly might be tedious.

Prime factorizing 2520, gives us  2^3 * 3^2 * 5^1 * 7^1 = (2^2)^1 * 2 * (3^2)^1 * 5^1 * 7^1

We use the counting rule on the powers of the perfect squares (the ones outside the brackets)

The number of perfect squares = (1 + 1) * (1 + 1) = 4

Option B

Arun Kumar

EgmatQuantExpert · Answer

Given

• A number 2520.

To Find

• The number of factors that are a perfect square.

Approach and Working Out

• 2520 =  2^3  ×  3^2  × 5 × 7
 o We need to manipulate it in a way that one part becomes a perfect square that is the highest possible one. 
• 2520 =  2^2  ×  3^2  × 2 × 5 × 7
 o We need to deal with  2^2  ×  3^2 . 
• Manipulating it further to make sure the term within parenthesis are squares ( 2^2 ) ^1  × ( 3^2 ) ^1 
 o Number of factors that are squares = (1 + 1) × (1 + 1) = 4

Correct Answer: Option B

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How many factors of the number 2520 are perfect squares?

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How many factors of the number 2520 are perfect squares?

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