mathquestionxtw

If 34x + 6 congruent 2 mod 20, 34x + 4 is divisible by 20.

We use trial and error(again) and test for each negative integer from -1.

34(-1) + 4 = -30 which is not divisible by 20.

34(-2) + 4 = -60<-- OMG answer already here. It is divisible by 20.

Therefore the largest negative integer which satisfies the congruence is -2 :)

~The smartest cookie in the world.

Isn't it -3?

Find the largest negative integer x which satisfies the congruence

$34x+6\equiv 2\pmod {20}$.

$x = -6+10n \quad | \quad n \in \mathbb{Z}$

The largest negative integer x = - 6

Check:

$\begin{array}{|rcll|} \hline && 34\cdot(-6)+6 \pmod {20} \\ &\equiv & -204 + 6 \pmod {20} \\ &\equiv & -198 \pmod {20} \\ &\equiv & -18 \pmod {20} \\ &\equiv & -18+20 \pmod {20} \\ &\equiv & 2 \pmod {20} \\ \hline \end{array}$