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Modular arithmetic

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Rule
What does $a \equiv r \pmod n$ mean?
Answer
$a \div n$ leaves remainder $r$ . For example $23 \equiv 2 \pmod 7$ , because $23 = 3 \times 7 + 2$ .
A modulo-5 clockOnly the remainder matters: where you stop
From the lesson: Modular arithmetic
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Which answers are allowed in modulo 7?
Answer
Only $0, 1, 2, \dots, 6$ . For a negative number, add 7 until it is in range: $-3 \equiv 4 \pmod 7$ .
From the lesson: Modular arithmetic
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Work out $5 \times 6 + 4$ in modulo 7.
Answer
Work it out as usual, then take the remainder on dividing by 7: $34 = 4 \times 7 + 6$ , so the answer is 6.
From the lesson: Modular arithmetic
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How do you solve $3x \equiv 2 \pmod 5$ ?
Answer
Try each value from 0 to 4 and keep the one that works: $3 \times 4 = 12 \equiv 2$ , so $x = 4$ .
From the lesson: Modular arithmetic
Which method?
WAEC 2019 · Paper 2 · Q1 (a)
Draw a table for multiplication $\otimes$ in modulo 8 on the set $T = \{2, 3, 5, 7\}$ .
How do you fill each cell of the table?
Answer
Multiply the two numbers, then take the remainder on dividing by 8. For example $5 \otimes 7 = 35 = 4 \times 8 + 3$ , so that cell is 3.
From the lesson: Modular arithmetic Try the question