How Many Odd Factors does 210 have?

Question: How many odd factors does 210 have?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

“How many odd factors does 210 have?” - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Approach 1

We have to list out all the factors of 210 and check which of them are the odd factors.

All the factors of 210 are -

1 * 210
2 * 105
3 * 70
5 * 42
6 * 35
7 * 30
10 * 21
14 * 15

Among these 1, 105, 3, 5, 35, 7, 21, 15 are the odd factors of 210.
Therefore we can see that there are 8 odd factors of 210 and hence E is the correct answer.

Approach 2

It is asked in the question how many odd factors does 210 have
210 = \((2^1 * 3^1 * 5^1 * 7^1)\)
Now there is a theory that says
The total number of factors of a number are the product of number of ways of choosing each prime factor.
If a number has n prime factors = a1, a2, a3……. an
Then the total number of factors will be = \(2^n\)
And odd factors will be the product of number of ways of choosing each odd prime factor.

Now coming back to the question,
210 has 4 prime factors = 2,3,7,5
Total factors = (1+1)(1+1)(1+1)(1+1) = 16
Odd factors =\( 2^3 \)= 8
The correct answer will be option E.

Approach 3

As per the problem statement: 210 = 2 * 3 * 5 * 7

Odd factors (exclude 2) = 3^1 * 5^1 * 7^1 ==> (1+1) * (1+1) * (1+1) ----> 2 * 2 * 2 = 8

Via this process, the even factors can also be found out -→

{Total factor} - {Odd Factors} ---> 16 - 8 = 8

Hence, E is the correct answer.

Correct Answer: E

Suggested GMAT Quant Questions

• 12 marbles are selected at random from a large collection of white, red, green and yellow Marbles. 
• The Set S of Numbers has the following Properties
• Two Varieties of Sugar are Mixed together in a Certain Ratio
• If a Carton Containing a Dozen Mirrors is Dropped which of the following Cannot be the Ratio of Broken Mirrors to Unbroken Mirrors?
• For How Many Values of k is 12^12 the Least Common Multiple of the Positive Integers 6^6, 8^8 and k?
• 4 Out Of 15 Apples Are Rotten. They Are Taken Out One By One At Random
• How Many Litres of a 90% Solution of Concentrated Acid Needs to be Mixed with a 75% Solution
• A Zookeeper Counted the Heads of the Animals in a Zoo and Found it to be 80
• If a Positive Integer n is Divided by 5, the Remainder is 3
• 2x+3y 
• The Number 2 - 0.5 is How Many Times the Number 1 - 0.5?
• A Clock Strikes 4 taking 9 seconds. In Order to Strike 12 at the Same Rate, the Time Taken is
• If 75 Percent of a Class Answered the First Question on a Certain Test Correctly
• In How Many Different Ways can 3 Identical Green Shirts and 3 Identical Red Shirts be Distributed Among 6 Children
• 4 Bells Toll Together at 9:00 A.M. They Toll After 7, 8, 11 and 12 seconds Respectively
• An Inlet Pipe can Fill in an Empty Cistern in 30 minutes Whereas a leak in the Bottom of the Cistern can Empty a Filled Tank in 40 minutes
• How Many Odd Factors does 210 have?
• If 80 Lamps Can Be Lighted, 5 Hours Per Day For 10 Days For $21.25
• A Circle with a Radius R is Inscribed into a Square with a Side K.
• Did Insurance Company K have More than $300 Million in Total Net Profits Last Year?