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HCF

by shashank.ism » Mon Feb 08, 2010 7:34 am
Two numbers have 36 factors each and HCF of these two numbers is 36. What is the minimum possible LCM of these two numbers if the power of any prime factor in these two
numbers is not more than 3?

a) cube(2) cube(3) sqr(5) 7
b) sqr(2) sqr( 3) cube(5) cube(7)
c) cube(2) cube( 3) sqr(5) ×sqr( 7)
d) cube(2) 32 cube(5) sqr(7)
e) cube(2) sqr(3) sqr(5) sqr(7)
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by firdaus117 » Sun Feb 21, 2010 8:19 am
Keeping in mind that the power of any prime factor in these two numbers is not more than 3,
36=4*3*3
Hence,the numbers are of the form of a^3b^2c^2.
As LCM is to be kept minimum,two numbers are 2^3*3^2*5^2 and 2^2*3^3*7^2.
The LCM of the two numbers will be cube(2) cube( 3) sqr(5) Xsqr( 7)
Option C :)
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by abhi332 » Sun Feb 21, 2010 9:44 am
I am confuse, How did you got "As LCM is to be kept minimum,two numbers are 2^3*3^2*5^2 and 2^2*3^3*7^2. "
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by ajith » Sun Feb 21, 2010 9:54 am
shashank.ism wrote:Two numbers have 36 factors each and HCF of these two numbers is 36. What is the minimum possible LCM of these two numbers if the power of any prime factor in these two
numbers is not more than 3?

a) cube(2) cube(3) sqr(5) 7
b) sqr(2) sqr( 3) cube(5) cube(7)
c) cube(2) cube( 3) sqr(5) ×sqr( 7)
d) cube(2) 32 cube(5) sqr(7)
e) cube(2) sqr(3) sqr(5) sqr(7)

36 = 2^2*3^2

The two numbers are 2^3*3^2*7^2, 2^2*3^3*5^2

LCM = cube(2) cube( 3) sqr(5) ×sqr( 7)

C
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by firdaus117 » Sun Feb 21, 2010 10:17 am
abhi332 wrote:I am confuse, How did you got "As LCM is to be kept minimum,two numbers are 2^3*3^2*5^2 and 2^2*3^3*7^2. "
For LCM to be minimised,we need to minimise the two numbers as well.Why?Because LCM itself is the number greater than or equal to the larger of the two number.
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by abhi332 » Sun Feb 21, 2010 10:46 am
if, 1st number = 2^3*3^2*5^2

why can't we take 2nd number same as first number i.e. 2^3*3^2*5^2 (since we want to minimize Lcm of the two numbers )

there is nothing in question given that two numbers are not equal.

Please correct me if I am missing anything.
Last edited by abhi332 on Sun Feb 21, 2010 11:35 am, edited 2 times in total.
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by firdaus117 » Sun Feb 21, 2010 11:29 am
abhi332 wrote:if, 1st number = 2^3*3^2*5^2

why can't we take 2nd number same as first number i.e. 2^3*3^2*5^2 (since we want to minimize Lcm of the two numbers )

there is nothing in question given that two numbers are not equal.

Please correct me if I am missing anything.
Ohh,they are different numbers because it is also given that the HCF is 36 which means if they would be equal they each will be equal to 36.But,36 has 9 roots only. :)
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