Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized for You

we will pick new questions that match your level based on your Timer History

Track Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 350,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

(1) The number of distinct factors of N is even. (2) The sum of all distinct factors of N is even.

SOL:

St1: Tip: If the number of distinct factors is even, then N cannot be a perfect square. The number of distinct factors of a perfect square is always odd. Remember this!! => SUFFICIENT

St2: Tip: If the sum of all distinct factors of N is even, then N cannot be a perfect square. The sum of all distinct factors of a perfect square is always odd. Remember this too!! => SUFFICIENT

How can you demonstrate the the sum of all the factor of a square is odd?

This question is from MGMAT.

Thanks

Well I knew it would come to this......this is going to be a lengthy explanation!

Any number N when expressed in terms of powers of its prime factors takes the form N = a^p * b^q * c^r ........... where a, b, c, ....... are prime numbers and p, q, r, ....... are the the number of powers. Eg: N = 72 = 2^3 * 3^2 Here a = 2, b = 3 & p = 3, q = 2

There is a formula for finding the sum of all the factors of N. It goes this way: S(Factors of N) = (a^(p+1) - 1) * (b^(q+1) - 1) * (c^(r+1) - 1) * ........ ( a - 1 ) * ( b - 1 ) * ( c - 1 ) * ........

Lets illustrate with the help of an eg why this formula will always yield an odd sum when N is a perfect square. Say N = 8100 = 2^2 * 3^4 * 5^2

(2-1), (3-1) & (5-1) will get canceled from the denominator as well as numerator.

Thus we have, S(Factors of 8100) = (2^2 + 2 + 1) * (3^4 + 3^3 + 3^2 + 3 + 1) * (5^2 + 5 + 1) All the 3 expressions above are odd numbers: (2^2 + 2 + 1) = E + E + O = O ....... Even nos + an odd number = odd (3^4 + 3^3 + 3^2 + 3 + 1) = O + O + O + O + O = O ....... Sum of an odd no of odd nos is odd (5^2 + 5 + 1) = O + O + O = O ...... Sum of an odd no of odd nos is odd where O = Odd & E = Even

Thus we get, S(Factors of 8100) = O * O * O = O

This will be the case for all perfect squares because perfect square will always have even number of powers of prime factors. => Thus S(N) formula will have even + 1 = odd number of powers. => For (a^n - b^n) where n is odd there will always be odd no of terms in the expansion as we saw above. => The sum of these odd no of odd terms will always be odd. Even in case of (2^odd - 1) there will a string of even nos + one odd no at the end thus making the whole expression odd. => Multiplication of these odd nos will always be odd.

Hence proved. Phew!!!!

PS: I wish the old multiple kudos system were still functional !!!! _________________

Kudos, guys, for the good question and explanation. My approach:

n = 2^a*3^b*5^c..... - any positive integer. N = (1+a)(1+b)(1+c) - number of distinct factors. to be perfect square, all a, b, c ... for N must be even.

1) The number of distinct factors of N is even. N = (1+even)(1+even)... - always odd.

(2) The sum of all distinct factors of N is even. the sum of even factors will be always even but if the number of odd factors is odd, the sum will be odd. Let's see what we have for perfect square: exclude any power of 2: The number of odd factors of N --> (1)(odd)(odd)... = odd So, we always have odd number of odd factors for a perfect square. Therefore, the sum of all factors will be also odd. _________________

Kudos, guys, for the good question and explanation. My approach:

n = 2^a*3^b*5^c..... - any positive integer. N = (1+a)(1+b)(1+c) - number of distinct factors. to be perfect square, all a, b, c ... for N must be even.

1) The number of distinct factors of N is even. N = (1+even)(1+even)... - always odd.

(2) The sum of all distinct factors of N is even. the sum of even factors will be always even but if the number of odd factors is odd, the sum will be odd. Let's see what we have for perfect square: exclude any power of 2: The number of odd factors of N --> (1)(odd)(odd)... = odd So, we always have odd number of odd factors for a perfect square. Therefore, the sum of all factors will be also odd.

Hi Walker, could you explain how you concluded that the number of odd factors is odd? _________________

How can you demonstrate the the sum of all the factor of a square is odd?

This question is from MGMAT.

Thanks

Well I knew it would come to this......this is going to be a lengthy explanation!

Any number N when expressed in terms of powers of its prime factors takes the form N = a^p * b^q * c^r ........... where a, b, c, ....... are prime numbers and p, q, r, ....... are the the number of powers. Eg: N = 72 = 2^3 * 3^2 .............. => Multiplication of these odd nos will always be odd.

Hence proved. Phew!!!!

PS: I wish the old multiple kudos system were still functional !!!!

Outstandingly done dude.... Could somebody tell me from where I should study Number Theory and also is there a source of good practice questions for number theory? Is thr some compilation of number theory questions present on gmatclub....like walker's compilation of probability???

Outstandingly done dude.... Could somebody tell me from where I should study Number Theory and also is there a source of good practice questions for number theory? Is thr some compilation of number theory questions present on gmatclub....like walker's compilation of probability???

You could start with Manhattan Strategy guide on Numbers for theory. _________________

Can some one help, what disintict factors means - would not it be 2,3,5 in case of 8100?

Walker - Can you please guide me how can I get your notes for probability theory.

Thanks guys this awesome forum, handsdown!!

- 2,3,5 are prime factors but the problem says all distinct factors: 1,2,3,5,6,10,15,18.... - You can see the link in my signature: Comb/Prom - it is a list of links to problems arranged with difficulty level. _________________

(1) The number of distinct factors of N is even. (2) The sum of all distinct factors of N is even.

n = 2^a*3^b*5^c..... for any positive integer. N = (1+a)(1+b)(1+c)...gives us the number of distinct factors. Prime factors: 2,3,5, 7, 11, 13....... Distinct factors: 1,2,3,5,6,10,15,18....

From Statement 1 Rule: The number of distinct factors of a perfect square is always [highlight]odd[/highlight]. E.g. n=144=(2^4)(3^2) ---> N=(1+4)(1+2)=[highlight]15[/highlight] Sufficient

From Statement 2 Rule: The sum of all distinct factors of a perfect square is always odd. Sum(Factors of N) = (a^(p+1) - 1) * (b^(q+1) - 1) * (c^(r+1) - 1) * ........ ( a - 1 ) * ( b - 1 ) * ( c - 1 ) * ........

(2-1), (3-1) & (5-1) will get canceled from the denominator as well as numerator.

Thus we have, S(Factors of 8100) = (2^2 + 2 + 1) * (3^4 + 3^3 + 3^2 + 3 + 1) * (5^2 + 5 + 1) All the 3 expressions above are odd numbers: (2^2 + 2 + 1) = E + E + O = O ....... Even nos + an odd number = odd (3^4 + 3^3 + 3^2 + 3 + 1) = O + O + O + O + O = O ....... Sum of an odd no of odd nos is odd (5^2 + 5 + 1) = O + O + O = O ...... Sum of an odd no of odd nos is odd where O = Odd & E = Even

Thus we get, S(Factors of 8100) = O * O * O = O

This will be the case for all perfect squares because perfect square will always have even number of powers of prime factors. => Thus S(N) formula will have even + 1 = odd number of powers. => For (a^n - b^n) where n is odd there will always be odd no of terms in the expansion as we saw above. => The sum of these odd no of odd terms will always be odd. Even in case of (2^odd - 1) there will a string of even nos + one odd no at the end thus making the whole expression odd. => Multiplication of these odd nos will always be odd.

+1 kudos to walker and samrus98! _________________

"The best day of your life is the one on which you decide your life is your own. No apologies or excuses. No one to lean on, rely on, or blame. The gift is yours - it is an amazing journey - and you alone are responsible for the quality of it. This is the day your life really begins." - Bob Moawab

: Social ventures, both non-profits and for-profits, seek to better the world in such industries as education, microfinance, workforce development, public health and community development, among others. Organizations that...

Essay B for Stanford GSB will essentially ask you to explain why you’re doing what you’re doing. Namely, the essay wants to know, A) why you’re seeking...

Over the last week my Facebook wall has been flooded with most positive, almost euphoric emotions: “End of a fantastic school year”, “What a life-changing year it’s been”, “My...