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Loser66
 one year ago
Show that if a, b and c are integers with c  ab, then c (a,c)(b, c)
Please, help
Loser66
 one year ago
Show that if a, b and c are integers with c  ab, then c (a,c)(b, c) Please, help

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Loser66
 one year ago
Best ResponseYou've already chosen the best response.1It is easy to prove by usual way: (a, c ) = xa + yc for some x, y in Z (b, c) = sb + tc for some s, t in Z (a,c) (b,c) = (xa + yc)(sb + tc) = xasb + xatc + ycsb + yctc = ab(xs) + c ( atx ) + c (ysb) + c ( ytc) c  ab for the first term , hence c divide the sum. Problem is : my Prof wants me to use prime factorization to prove it.

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1My attempt: \(c = \prod p_i^{m_i}\) \(ab= \prod p_i^{s_i}\) \(cab \implies m_i < s_i\) Then, no where to go. :(

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I've only tried the easy way...interesting how you would use it for prime factorisation

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1I don't get why c  ab, implies \(c_i \geq a_i + b_i\)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2Oops, it is a typo, fixed here : Let : \(a = \prod p_i^{a_i}\) \(b = \prod p_i^{b_i}\) \(c = \prod p_i^{c_i}\) then, \((a,c) = \prod p_i^{\min\{a_i,c_i\}}\) \((b,c) = \prod p_i^{\min\{b_i,c_i\}}\) \(c\mid ab \implies c_i \le a_i+b_i\) so, \(\min\{a_i,c_i\}+\min\{b_i,c_i\} \le a_i+b_i \ge c_i \blacksquare \)

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1Got you. Thank you so much.
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