Calculate $40*9 \pmod{23}.$ Express your answer as a non-negative integer that is less than $23$.

1 answer

To find \(40 \times 9 \pmod{23}\), we first compute \(40 \mod 23\).

\[
40 \div 23 = 1.739
\]

The integer part of the division is 1, so we calculate the remainder:

\[
40 - 23 \times 1 = 40 - 23 = 17
\]

Thus,

\[
40 \equiv 17 \pmod{23}
\]

Next, we need to find \(17 \times 9 \pmod{23}\). First, calculate \(17 \times 9\):

\[
17 \times 9 = 153
\]

Next, we find \(153 \mod 23\):

\[
153 \div 23 = 6.652
\]

The integer part of the division is 6, so we calculate the remainder:

\[
153 - 23 \times 6 = 153 - 138 = 15
\]

Thus,

\[
153 \equiv 15 \pmod{23}
\]

Therefore, \(40 \times 9 \pmod{23}\) is:

\[
\boxed{15}
\]