Question

modular arithmetic - reduction then evaluation
report all 4 of your answers on the clock.
reduce each of the following.
52 ≡ \boxed{} (mod 4)
74 ≡ \boxed{} (mod 6)
96 ≡ \boxed{} (mod 11)
evaluate 21 + 38 - 46 × 21 ≡ \boxed{} (mod 19)

Explanation:

Step1: Solve \(52 \equiv \square \pmod{4}\)

To find \(52 \bmod 4\), we divide 52 by 4. Since \(52\div4 = 13\) with a remainder of 0, \(52 \equiv 0 \pmod{4}\).

Step2: Solve \(74 \equiv \square \pmod{6}\)

Divide 74 by 6. \(6\times12 = 72\), so \(74 - 72 = 2\). Thus, \(74 \equiv 2 \pmod{6}\).

Step3: Solve \(96 \equiv \square \pmod{11}\)

Divide 96 by 11. \(11\times8 = 88\), and \(96 - 88 = 8\). So, \(96 \equiv 8 \pmod{11}\).

Step4: Evaluate \(21 + 38 - 46 \times 21 \equiv \square \pmod{19}\)

First, reduce each number modulo 19:

• \(21 \bmod 19 = 2\)
• \(38 \bmod 19 = 0\)
• \(46 \bmod 19\): \(19\times2 = 38\), \(46 - 38 = 8\), so \(46 \equiv 8 \pmod{19}\)
• \(21 \bmod 19 = 2\) (again for the last 21)

Now substitute into the expression: \(2 + 0 - (8 \times 2) \pmod{19}\)
Calculate \(8 \times 2 = 16\)
Then \(2 + 0 - 16 = -14\). To make this positive modulo 19, add 19: \(-14 + 19 = 5\). So \(21 + 38 - 46 \times 21 \equiv 5 \pmod{19}\).

Answer:

For \(52 \equiv \square \pmod{4}\), the answer is \(0\).
For \(74 \equiv \square \pmod{6}\), the answer is \(2\).
For \(96 \equiv \square \pmod{11}\), the answer is \(8\).
For \(21 + 38 - 46 \times 21 \equiv \square \pmod{19}\), the answer is \(5\).