(a) Copy and complete the following table for multiplication modulo 11. \(\otimes\) 1 5 9 10 1 1 5 9 10 5 5 9 9 10 10 Use the table to : (i) evaluate \((9 \...

Answer Details

(a) Multiplication table modulo 11 (multiply, then take the remainder on division by 11).

For example \(5\otimes 5 = 25 = 2(11)+3 \equiv 3\), \(9\otimes 9 = 81 = 7(11)+4 \equiv 4\), \(10\otimes 10 = 100 = 9(11)+1 \equiv 1\).

\(\otimes\)	1	5	9	10
1	1	5	9	10
5	5	3	1	6
9	9	1	4	2
10	10	6	2	1

(i) \((9\otimes 5)\otimes(10\otimes 10) = 1 \otimes 1 = \mathbf{1}\).

(ii)(1) \(10\otimes m = 2\): from the row for 10, \(10\otimes 9 = 2\), so the truth set is \(\{9\}\).

(ii)(2) \(n\otimes n = 4\): scanning the leading diagonal, only \(9\otimes 9 = 4\), so the truth set is \(\{9\}\).

(b) The fraction. In lowest terms it equals \(\tfrac{3}{4}\), so let it be \(\dfrac{3k}{4k}\).

Numerator doubled is 34 more than the denominator:

\[2(3k) = 4k + 34 \;\Rightarrow\; 6k - 4k = 34 \;\Rightarrow\; 2k = 34 \;\Rightarrow\; k = 17\] \[\text{Fraction} = \frac{3(17)}{4(17)} = \frac{51}{68}\]

Check: \(2(51) = 102 = 68 + 34\). The fraction is \(\mathbf{\tfrac{51}{68}}\).