設 a、b、c 為正整數，求 a＋b－c 之值為何？

Question

設 a、b、c 為正整數，且 (125x343)2x(77x55)4/3855 = 5a x 7b x11c，求 a＋b－c 之值為何？

Answer 1

∵ (125×343)^2×(77×55)^4/385^5
=(5^3×7^3)^2×(7×5×11^2)^4/(5×7×11)^5
=(5^10×7^10×11^8)/ (5^5×7^5×11^5)
=5^5×7^5×11^3
∴a=5,b=5,c=3
→a+b－c=7＃

Answer 2

(１２３＊３４３)^2＊(７７＊５５)^4／３８５^5
=[(５^3＊７^3)^2]＊[(１１＊７＊５＊１１)^4]／[(５＊７＊１１)^5]
=５^5＊７^6＊１１^4＊７^4＊５^4＊１１^4／(５^5＊７^5＊１１^5)
=(５^6)＊[７^6]＊{１１^4}＊[７^4]＊(５^4)＊{１１^4}
／(５^5)／[７^5]／{１１^5}
=(５^6)＊(５^4)／(５^5)
＊[７^6]＊[７^4]／[７^5]
＊{１１^4}＊{１１^4}／{１１^5}
=５^5＊７^5＊１１^3
得a=５
　b=５
　c=３
ａ＋ｂ－ｃ＝５＋５－３＝７

Answer 3

(125x343)2x(77x55)4/3855 =(53x73)2x(7x5x112)4/(5x7x11)5=510x710x118/55x75x115=55x75x113= 5a x 7b x11ca=5;b=5;c=3a+b-c=5+5-3=7

Answer 4

(125x343)2x(77x55)4/3855 = 5a x 7b x11c=> (53x73)2x(7x11x5x11)4/(5x7x11)5 = 5a x 7b x11c=> (510x710x118)/(55x75x115) = 5a x 7b x11c=> 55x75x113 = 5a x 7b x11c=> a=5, b=5, c=3所以 a+b-c=5+5-3=7

Answer 5

(125*343)^2 * (77*55)^4 / 385^5
=(5^3 * 7^3)^2 * (7*11*5*11)^4 / (5*7*11)^5
=(5^6 * 7^6 * 7^4 * 11^4 * 5^4 * 11^4) / (5^5 * 7^5 * 11^5)
=5^(6+4-5) * 7^(6+4-5) * 11^(4+4-5)
=5^5 * 7^5 * 11^3

a=5 , b=5 , c=3
a+b-c= 7