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*5^2Total 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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65% (01:34) correct
