How many ordered pairs of positive integers (m,n) are there such that the least common multiple of m and n is 2^3 7^4 13^13?
parheliubdr
Answered question
2023-02-20
Answer & Explanation
Camuccinirk84
Beginner2023-02-21Added 4 answers
So m=2^a_1 7^b_1 13^_c1 and n=2^a_2 7^a_2 13^a_2
For some non-negative integers a1,b1,c1,a2,b2,c2.
2^3 y^4 13^13 is the least common multiple.
max {a1,a2}=3
max {b1,b2}=4
max {c1,c2}=13
{a1a2} can be equal to
(0,3),(1,3),(2,3),(3,3),(3,2),(3,1) or 3,0) a) total of 7 choices.
(b1,b2)=4 for {b1,b2} we have 2*4+1=9 Choices.
Maximum {c1,c2}=13, for {c1,c2} we have 2*13+1=27 choices.
And the number of ordered pairs (m,n)=7*9*27=1701
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