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Paired number GCD and LCM (Posted on 2019-05-03) Difficulty: 4 of 5
How many ordered pairs of positive integers satisfy

gcd(m3, n2) = 22*32
lcm(m2, n3) = 24*34*56 ?

No Solution Yet Submitted by Danish Ahmed Khan    
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solution Comment 1 of 1
1) gcd(m^3,n^2)=2^2*3^2

2) lcm(m^2,n^3)=2^4*3^4*5^6


from 1 we can deduce that both m,n are divisible by 2 and 3.
from 2 we know that one of them is divisible by 5, however it can't
be both because otherwise 1 would have a factor of 5.

let

m=2^a*3^b*5^i
n=2^c*3^d*5^j

where either i=3,j=0 or i=0,j=2 to reflect which one has the factor of 5

from 1 we get
min(3a,2c)=2
min(3b,2d)=2
since 3 does not divide 2 we know that the min value comes from 2c,2d thus
c=1,d=1

from 2 we get
max(2a,3)=4
max(2b,3)=4
so from this we can deduce a=b=2


thus we have 2 solutions
1) m=2^2*3^2*5^3 and n=2^1*3^1
2) m=2^2*3^2 and n=2^1*3^1*5^2

  Posted by Daniel on 2019-05-03 20:12:45
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