推 AZsorcerer :哈 我用x也是找不到其他結果 06/08 19:34
→ AZsorcerer :應該說就是多8833那個 06/08 19:35
※ 編輯: JohnMash 來自: 27.147.57.77 (06/09 19:12)
※ 引述《mathtype (ψ 綻 藍 )》之銘言：
: 12的平方 + 33的平方 = 1233
: 若限定和的結果為四位數,則類似的情況有哪些?
y^2-y=100x-x^2=-(x-50)^2+2500<=2500
hence, y^2<=2500+y<=2599<51^2
9<y<51
x^2+y^2=100x+y
x^2+y^2=x+y mod99
x(x-1)+y(y-1)=0 mod9
x(x-1)+y(y-1)=0 mod11
since y(y-1) is even, x must be even,
then 4|y(y-1)
0*1=0,1*2=2,2*3=6,3*4=3,4*5=2,5*6=3,6*7=6,7*8=2 mod9
hence, y=0,1,3,4,6,7 mod9
then y=12,13,16,
21,24,25,28,
33,36,37,
41,45,48,49
0*1=0,1*2=2,2*3=6,3*4=1,4*5=9,5*6=8,6*7=9,7*8=1,8*9=6,9*10=2 mod11
hence, y=0,1,2,5,7,10 mod11
then y=12,13,16,
21,24,29,
32,33,
40,44,45,49
possible y=12,13,16,21,24,33,45,49
and y(y-1) should be factored to a*(100-a)
y(y-1)=12*11 NO
=13*12 NO
=16*15=2*120=3*80 NO
=21*20=2*210=3*140=4*105=5*84 NO
=24*23=4*138=6*92 NO
=33*32=8*132=12*88 YES
=45*44=20*99=22*90=30*66 NO
=49*48=42*56=28*84 NO
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