What is an easy, efficient way to do these sort of problems?
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Originally Posted by dwsmith What is an easy, efficient way to do these sort of problems? and Also and So here's the prime factors we're interested in and the appropriate powers: So to find an and , just choose an exponent from each row to assign to and choose the other for . For example we could take , which forces .
What do you mean by just choose an exponent?
Originally Posted by dwsmith What do you mean by just choose an exponent? Choose an exponent for each row, either from the min column or the max column.
Originally Posted by chiph588@ Choose an exponent for each row, either from the min column or the max column. so there are 5 to try?
Originally Posted by dwsmith so there are 5 to try? Yes, for every prime dividing the gcd or lcm. In this case we have 4 primes: 2,3,5,7.
Then there are 5*4=20 possibilities to try.
Originally Posted by dwsmith Then there are 5*4=20 possibilities to try. No there are solutions.
Originally Posted by chiph588@ No there are solutions. Well the exponent of prime factor 5 for both a and b is always 1, so it's actually distinct pairs (a,b). Note that this number counts (420, 40) and (40, 420) as distinct.
Originally Posted by chiph588@ No there are solutions. Are the possible solutions always can to be 2^{number of primes}?
Originally Posted by dwsmith Are the possible solutions always can to be 2^{number of primes}? Let a be the number of primes where . Then the number of solutions is .
Is there a way to do permutations of the exponents to solve all the solutions or must I just plug them in?
Originally Posted by dwsmith Is there a way to do permutations of the exponents to solve all the solutions or must just plug them in? All possible permutations form all the solutions.
Here is another solution. Since , you can write and , where and are relatively prime. Beacuse , and then . Now choose such that . There are 8 possible ways.
Originally Posted by melese Here is another solution. Since , you can write and , where and are relatively prime. Beacuse , and then . Now choose such that . There are 8 possible ways. I see what you are doing but can you expand your example?