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k = 2^a3^b5^c

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k = 2^a3^b5^c

by sanju09 » Tue Sep 14, 2010 4:31 am
k is a positive integer and 225 and 216 are both divisors of k. If k = 2^a3^b5^c, where a, b and c are positive integers, what is the least possible value of a + b + c?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8



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by ankur.agrawal » Tue Sep 14, 2010 4:40 am
E- 8

225=5^2 * 3^2

216= 3^3 * 2^3

LCM of 225 & 216 = 2^3 * 3^3 * 5^2

so a=3, b=3 & c= 2

a+b+c = 8
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by sanju09 » Tue Sep 14, 2010 4:43 am
ankur.agrawal wrote:E- 8

225=5^2 * 3^2

216= 3^3 * 2^3

LCM of 225 & 216 = 2^3 * 3^3 * 5^2

so a=3, b=3 & c= 2

a+b+c = 8
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by Maciek » Tue Sep 14, 2010 4:56 am
Sanju09!

IMO E

225 = 3*3*5*5 = 3^2*5^2
216 = 2*2*2*3*3*3 = 2^3*3^3

Formula for least common multiple( LCM) of A and B:
LCM(A,B) = |A*B|/GCD(A,B)

greatest common divisor( GCD) of 225 and 216 is
GCD(225,216) = 3*3 = 3^2

LCM(225,216) = |225*216|/GCD(225,216) = |3^2*5^2*2^3*3^3|/3^2 = 2^3*3^3*5^2
k = 2^a3^b5^c = LCM(225,216) = 2^3*3^3*5^2
a = 3, b = 3, c = 2
a + b + c = 8

Hope it helps!
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Maciek
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