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Determine the smallest non-negative integer  that satisfies the congruences:

 

 

 

                                                                                                   \(\begin{align*} &a\equiv 2\pmod 3,\\ &a\equiv 4\pmod 5,\\ &a\equiv 6\pmod 7,\\ &a\equiv 8\pmod 9. \end{align*}\)

 Jul 15, 2020
 #1
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The smallest n that works is 944.

 Jul 16, 2020
 #2
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Using Chinese Remainder Theorem with Modular Multiplicative Inverse which are incorporated into this short computer code, we get:

 

i=0;j=0;m=0;t=0;a=(3, 5,7, 9);r= (2, 4, 6, 8);c=lcm(a); d=c / a[i];n=d % a[i] ;loop1:m++; if(n*m % a[i] ==1, goto loop, goto loop1);loop:s=(c/a[i]*r[j]*m);i++;j++;t=t+s;m=0;if(i< count a, goto4,m=m);printc,"m + ",t % c;return

 

 

315 m +  314, where m =0, 1, 2, 3.......etc.

 

Then the smallest positive integer = 314

 Jul 16, 2020

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