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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k = 2^a3^b5^c
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sanju09
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ankur.agrawal
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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
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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sanju09
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kewlankur.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
The mind is everything. What you think you become. -Lord Buddha
Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001
www.manyagroup.com
Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001
www.manyagroup.com
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Maciek
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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!
Best,
Maciek
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!
Best,
Maciek
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