Hint Use the Chinese Remainder TheoremSolution x214x30x214x4

Hint: Use the Chinese Remainder Theorem

Solution

x²14x+30=x²14x+4919 = (x7)²19.

So the congruence is now (x7)²19 mod 1615

(x7)²19mod1615. However, 1615 is actually a multiple of 19, whence we split it into congruences via

Chinese Remainder Theorem,

1615=19175. Call y=x7   (just a shift of number, we can change it back later), then:

y² 19 mod 19

y²    19 (=2) mod 17

y²    19 (=4) mod 5

Need simultaneous solving. Each of these can be solved, and the solutions are:

y 0 mod 19

y ±6 mod 17

y ±2 mod 5

Now that we have these congruences, we combine them using the Chinese remainder theorem: The second two combine to give y 23/28/57/62 mod 85. Note that 57 already divides 19, but the rest need to be augmented with a suitable multiple of 19 to work. and results are : y ±57, ±703 mod 1615

To verify:

57²19=3230=21615

703²19=494190=3061615

Of course, we have to put back the resulting 7 shift we made earlier, and the results are:

x= 64/710/919/1565 .