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If a and b are positive integers such that gcd(a,b)=210, lcm[a,b]=210^4, and a < b, how many possible values are there for a?

 Feb 15, 2022
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i=0;a=1; c=210 *a^3;d=int(1944810000 / a^3); if(gcd(c, d)==210 and lcm(c,d)==210^4, goto5, goto7);i++;printi,"=",(c,d); a++;if(a<250, goto2, 0)

 

1 = (210, 1944810000)
2 = (1680, 243101250)
3 = (5670, 72030000)
4 = (26250, 15558480)
5 = (45360, 9003750)
6 = (72030, 5670000)
7 = (210000, 1944810)
8 = (576240, 708750)
9 = (708750, 576240)
10 = (1944810, 210000)
11 = (5670000, 72030)
12 = (9003750, 45360)
13 = (15558480, 26250)
14 = (72030000, 5670)
15 = (243101250, 1680)
16 = (1944810000, 210)
Note: You are only interested in the first 8 because a < b. You may drop the other 8 [from 9 to 16].

 Feb 16, 2022

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