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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 !!!! _________________
(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
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. _________________
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. _________________
(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! _________________
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