How many of the factors of 210 are odd numbers greater than 1?

Question

How many of the factors of 210 are odd numbers greater than 1? A. Three B. Four C. Five D. Six E. Seven 2024-01-30_01-03-36.png

Bunuel · Accepted Answer

Firstly, perform the prime factorization of 210:

210 = 2*3*5*7

To get only the odd factors, we ignore 2, and calculate the factors of  3*5*7 , which is  (1+1)(1+1)(1+1)=8 . However, this count includes 1 as a factor. So, the number of factors of 210 that are odd and greater than 1 is 7.

Answer: E.

P.S  Finding the Number of Factors of an Integer 
 
First, make the 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 respective 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 
 
The total number of factors of 450, including 1 and 450 itself, is  (1+1)(2+1)(2+1) = 2*3*3 = 18  factors.

gmatophobia · Answer

210 = 21 * 10 = 7 * 3 * 2 * 5

Number of odd factos = 2 * 2 * 2 = 8

This also includes 1. We have to remove that case.

Hence, 8 - 1 = 7

Option E

sachi-in · Answer

210 = 3 * 7 * 5 * 2

Only 3 , 7 , 5 can be used to form the odd factors

Now we can choose either 0 or 1 as power of each of 3 , 7 , 5, hence total 2 * 2 * 2 factors

Of the 8 factors one will be 3^0 * 5^0 * 7^0 = 1 which must be excluded being less than 1.

Hence total 8 - 1 = 7 odd factors greater than 1

Danou · Answer

Could you kindly elaborate on the steps following finding 3,5,7. Not sure I understand where 2x2x2 comes from?

Bunuel · Answer

Finding the Number of Factors of an Integer

First, make the 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 respective 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

The total number of factors of 450, including 1 and 450 itself, is  (1+1)(2+1)(2+1) = 2*3*3 = 18  factors.

_____________________________

Thus, the number of all positive factors of 210, which is 2 * 3 * 5 * 7, is (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 16 (notice that all the powers of the primes are 1).

Now, to find the odd factors of 210, we just ignore the powers of 2, so we basically need to find the number of factors of 3 * 5 * 7, which is (1 + 1)(1 + 1)(1 + 1) = 8. This, however, would include 1, and since we need the number of odd factors greater than 1, we get 8 - 1 = 7.

Hope it's clear.

2. Properties of Integers 
     Theory   
 Math: Number Theory 
 Number Properties: Tips and hints 
 Advanced Number Properties on GMAT (All 5 parts) 
 Do you make these 3 mistakes in GMAT Even-Odd Questions? 
 Everything about Factorials on the GMAT 
 Solving Complex Problems Using Number Line 
     Questions   
 Tough New Set: Number Properties Set 
 Power of a Number in a Factorial Problems 
 Number Properties Ability Quiz 
 Special Numbers and Sequences Problems 
 The E-GMAT Number Properties Knockout: 10 Great 700+ Ques for you! 
 Trailing Zeros Problems 
 DS Questions 
 PS Questions

For other subjects:
 ALL YOU NEED FOR QUANT ! ! ! 
 Ultimate GMAT Quantitative Megathread

Hope this helps.

GMATinsight · Answer

210 = 2*3*5*7 Odd factors of 210 will be the factors made of all odd prime factors of 210 i.e. Odd factors of 210 = Factors of 3*5*7 : CONCEPT : If N = a^p*b^q*c^r... where a, b, c... are distinct primes Number of factors of N = (p+1)*(q+1)*(r+1)*... i.e. Factors of 3*5*7 = (1+1)*(1+1)*(1+1) = 8 But one of these factors is 1 hence factors of 210 which are greater than 1 = 8-1 = 7 Answer: Option E --- GMATinsight | www.GMATinsight.com GMAT FOCUS PREP Expert Youtube: www.Youtube.com/GMATinsight

Danou · Answer

Thank you all, much clearer now!

bumpbot · Answer

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How many of the factors of 210 are odd numbers greater than 1?

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How many of the factors of 210 are odd numbers greater than 1?

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