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A number n( positive integer) when factorised can be written [#permalink]
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16 Nov 2005, 07:49
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A number n( positive integer) when factorised can be written as a^4*b^3*c^7.find number of perfect squares which are factors of n.a,b,c are prime >2.



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Re: number properties. [#permalink]
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16 Nov 2005, 08:06
laxieqv wrote: A number n( positive integer) when factorised can be written as a^4*b^3*c^7.find number of perfect squares which are factors of n.a,b,c are prime >2.
I got 17
a^2
a^4
b^2
c^2
c^4
c^6
(ab)^2
(bc)^2
(ac)^2
(a^2c^2)^2
(a^3c^3)^2
(ac^2)^2
(ac^3)^2
(a^2c^3)^2
(a^3c)^2
(a^3c^2)^2
(abc)^2
did I miss any.
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Re: number properties. [#permalink]
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16 Nov 2005, 08:23
laxieqv wrote: find number of perfect squares which are also factors of n, a, b, and c are prime >2.
hmmm........... looks like you prepared/made this question. your question is little unclear. i am assuming it as "find the perfect squares that are also factors of n, a, b, and c and assuming that n, a, b, and c are different primes and are greater than 2".
a^4 can be selected in two ways i.e. a^2 and a^4 (assuming a^2 and a^2 in a^4 are same) = 2 ways
b^3 can be selected in 1 way i.e. b^2= 1 way
c^7 can be selected in 4 ways i.e. c^2, c^4 and c^6= 3 ways
so the list = 2x1x3= 6 ways
a^2 b^2 c^2
a^2 b^2 c^4
a^2 b^2 c^6
a^4 b^2 c^2
a^4 b^2 c^4
a^4 b^2 c^6



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Re: number properties. [#permalink]
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16 Nov 2005, 09:55
laxieqv wrote: A number n( positive integer) when factorised can be written as a^4*b^3*c^7.find number of perfect squares which are factors of n.a,b,c are prime >2.
n = a^4 * b^3 * c^7
a^4 has 4/2=2 perfect squares
b^3 has 3/2=1 perfect squares
c^7 has 7/2=3 perfect squares
a^4*b^3 has min(4,3)/2 = 3/2 = 1 perfect square
b^3*c^7 has min(3,7)/2 = 3/2 = 1 perfect square
c^7*a^4 has min(7,4)/2 = 4/2 = 2 perfect squares
a^4*b^3*c^7 has min(4,3,7)/2 = 3/2=1 perfect square
total: 11
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hmm
I would say 6...but then 11 makes sense..
if N has perfect square factors...
they out to be: a^2.b^2.c^2 or a^4.b^2.c^4 or a^2.b^2.c^6..and so on.... i dont think we can count a^2.c^2.b^1 as a perfect square factor?...if we consider a^2*1 as a factor than this is a perfect square..b^2*1 is a perfect square factor...if we have a^2.b^2*1 also a factor etc...then 11 makes sense...



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Re: number properties. [#permalink]
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17 Nov 2005, 21:57
laxieqv wrote: A number n( positive integer) when factorised can be written as a^4*b^3*c^7.find number of perfect squares which are factors of n.a,b,c are prime >2.
laxi, where r u? probably busy in dating, hanmmm.....? donot forget that you have posted the question and have not answered/clearified yet.



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Re: number properties. [#permalink]
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17 Nov 2005, 22:14
nakib77 wrote: laxieqv wrote: A number n( positive integer) when factorised can be written as a^4*b^3*c^7.find number of perfect squares which are factors of n.a,b,c are prime >2. I got 17 a^2 a^4 b^2 c^2 c^4 c^6 (ab)^2 (bc)^2 (ac)^2 (a^2c^2)^2 (a^3c^3)^2 (ac^2)^2 (ac^3)^2 (a^2c^3)^2 (a^3c)^2 (a^3c^2)^2 (abc)^2 did I miss any.
buddy, some are not correct, for example: (a^3*c^3)^2 coz then we have a^6 ( out of range)



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Re: number properties. [#permalink]
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17 Nov 2005, 22:18
HIMALAYA wrote: laxieqv wrote: A number n( positive integer) when factorised can be written as a^4*b^3*c^7.find number of perfect squares which are factors of n.a,b,c are prime >2. laxi, where r u? probably busy in dating, hanmmm.....? donot forget that you have posted the question and have not answered/clearified yet.
hey, are you still thinking that i'm the one in my avatar, a happy gal with her BF?!!! ...my avatar is from a movie!!
BTW, Duttsit's approach is the best one



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Re: number properties. [#permalink]
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17 Nov 2005, 22:22
laxieqv wrote: HIMALAYA wrote: laxieqv wrote: A number n( positive integer) when factorised can be written as a^4*b^3*c^7.find number of perfect squares which are factors of n.a,b,c are prime >2. laxi, where r u? probably busy in dating, hanmmm.....? donot forget that you have posted the question and have not answered/clearified yet. hey, are you still thinking that i'm the one in my avatar, a happy gal with her BF?!!! ...my avatar is from a movie!! BTW, Duttsit's approach is the best one
ok, i understoo/understand in that way too.



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Re: number properties. [#permalink]
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19 Nov 2005, 07:04
duttsit wrote: laxieqv wrote: A number n( positive integer) when factorised can be written as a^4*b^3*c^7.find number of perfect squares which are factors of n.a,b,c are prime >2. n = a^4 * b^3 * c^7 a^4 has 4/2=2 perfect squares b^3 has 3/2=1 perfect squares c^7 has 7/2=3 perfect squares a^4*b^3 has min(4,3)/2 = 3/2 = 1 perfect square b^3*c^7 has min(3,7)/2 = 3/2 = 1 perfect square c^7*a^4 has min(7,4)/2 = 4/2 = 2 perfect squares a^4*b^3*c^7 has min(4,3,7)/2 = 3/2=1 perfect square total: 11
OH MY GOD! Where have I come?????? How can you guys diligently solve such problems?? I cant even comprehend what such a problem means many a times!
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rahul..duttsit approach is very clear...if you cant follow it...try strenghthning your number properties skills...get your self a MGMAT number properties book...
basically the rule. is.prime^even is always a perfect square so we know that that N has prime factors a,b and c...so now all even powers of these factors would constitute a perfect square..



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Hmmm let's see if this thinking is wrong.
For a we have 3 possibilities, a^0, a^2, a^4
For b we have 2 possibilities, b^0, b^2
For c we have 4 possibilities, c^0, c^2, c^4, c^6
So we would have 3*2*4=24 possibilities. The smallest of the perfect square factors would be 1, and the largest would be a^4b^2c^6.
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HongHu wrote: Hmmm let's see if this thinking is wrong. For a we have 3 possibilities, a^0, a^2, a^4 For b we have 2 possibilities, b^0, b^2 For c we have 4 possibilities, c^0, c^2, c^4, c^6
So we would have 3*2*4=24 possibilities. The smallest of the perfect square factors would be 1, and the largest would be a^4b^2c^6. a, b, and c, all, are prime and >2. laxieqv wrote: A number n( positive integer) when factorised can be written as a^4*b^3*c^7.find number of perfect squares which are factors of n.a,b,c are prime >2.
Last edited by HIMALAYA on 20 Nov 2005, 22:36, edited 1 time in total.



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Well I understood he wants factors of n. And he says that a, b, c are primes that are greater than 2. Did I miss something? :hmm:
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HongHu wrote: Well I understood he wants factors of n. And he says that a, b, c are primes that are greater than 2. Did I miss something? :hmm: i think you are ok. i have been reading . (full stop) as , (coma). laxieqv wrote: A number n( positive integer) when factorised can be written as a^4*b^3*c^7. find number of perfect squares which are factors of n. a,b,c are prime >2.



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Here's what I got for this problem:
different possible square factors:
a^2,a^4,b^2,c^2,c^4,c^6
Now for combinations of the above you can select one or less from each group therefore from the a group you can select none, a^2 or a^4  3 ways. from the b group you can select none or b^2  2 ways and from the c group you can select none, c^2, c^4 or c^6  4 ways.
Therefore total number of perfect squares that can be formed from the above are = 3*2*4 = 24 from which we will have to subtract one since one possibility was selecting none from each group. Therefore answer = 23.



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angels624 wrote: Therefore total number of perfect squares that can be formed from the above are = 3*2*4 = 24 from which we will have to subtract one since one possibility was selecting none from each group. Therefore answer = 23.
does the bold part happen? ..i don't think so. Or do you mean the case of a^0 * b^0 * c^0 (= 1)? If so, this case should still be counted coz 1 is considered a perfect square. Btw, we all overlooked the case of a^0, b^0 and c^0 , which was brightly pointed out by sis Honghu










