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I'm working a problem where I need to find the greatest common divisor and the least common multiple for integer pairs. I then need to express the gcd as a linear combination. I could use a sanity check to make sure I'm on the right track.

I need to find the gcd and lcm of the integer pair: 175 and 35

By visual inspection, I "see" the answers, but working it outâ€¦

The gcd: [tex] (175,35) = \prod_i p_i^{min(\alpha_i,\beta_i)}[/tex]

[tex] 175 = 5^1\cdot35 = 5^1\cdot5^1\cdot7^1 = 5^2\cdot7^1[/tex]

[tex] 35 = 5^1\cdot7^1[/tex]

[tex] (175,35) = 5^1\cdot7^1 = 35[/tex]

The lcm: [tex] [175,35] = \prod_i p_i^{max(\alpha_i,\beta_i)}[/tex]

[tex] [175,35] = 5^2\cdot7^1 = 175[/tex]

So now I need to express the gcd as a linear combination. Is this correct:

[tex]ax + by = d[/tex] where [tex]d = (a,b)[/tex]

[tex]175x + 35y = 35[/tex]

[tex]x = 0[/tex] and [tex]y = 1[/tex]

[tex]\therefore 175\cdot(0) + 35\cdot(1) = 35[/tex]

Hopefully I'm on the right track, although I can see there being multiple solutions for the linear combination.

Thanks in advance for your help, comments, tips, clues, commentaries...

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# Abtract Algebra Sanity Check

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