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Find the smallest positive integer n such that 617n= 943n (mod 18)

 Jul 6, 2019
 #1
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a=1; b=1;c=  617*a%18; d=943*b%18; if(c==d, goto5, goto6);printc, d, a, b; a++;if(a<100, goto2, 0);a=1;b++;if(b<100, goto2, discard=0;

 

The smallest positive integer n = 9

 Jul 6, 2019
 #2
avatar+6046 
0

\(617n=943n \pmod{18}\\ -326n = 0 \pmod{18}\\ -326n + 342n = 0 \pmod{18}\\ 16n = 0 \pmod{18}\\ \text{now we look for the least common multiple of 16 and 18}\\ LCM(16,18)=144 = 16\cdot 9\\ n=9\)

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 Jul 7, 2019

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