enigma123 wrote:

How many factors does 36^2 have?

2 8 24 25 26

Answer is 25 i.e. D. Can someone please explain how?

Finding the Number of Factors of an IntegerFirst make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\).

NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.

For more on number properties check:

math-number-theory-88376.htmlBACK TO THE ORIGINAL QUESTION:

How many factors does 36^2 have?A. 2

B. 8

C. 24

D. 25

E. 26

\(36^2=(2^2*3^2)^2=2^4*3^4\) and according to above it'll have (4+1)(4+1)=25 different positve factros, including 1 and 36^2 itself.

Answer: D.

5 seconds approach: 36^2 is a perfect square. # of factors of perfect square is always odd (as perfect square has even powers of its primes then when adding 1 to each and multiplying them as in above formula you'll get the multiplication of odd numbers which is odd). Only answer choice D is odd thus it must be correct.

Answer: D.

Tips about the perfect square:

1. The

number of distinct factors of a perfect square is ALWAYS ODD.

2. The

sum of distinct factors of a perfect square is ALWAYS ODD.

3. A perfect square ALWAYS has an

ODD number of Odd-factors, and

EVEN number of Even-factors.

4. Perfect square always has

even powers of its prime factors.

Hope it helps.

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