(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 otimes 5) otimes (10 otimes 10));
(ii) find the truth set of :(1) (10 otimes m = 2); (2) (n otimes n = 4)
(b) When a fraction is reduced to its lowest term, it is equal to (frac{3}{4}). The numerator of the fraction when doubled would be 34 greater than the denominator. Find the fraction.
Explanation
(a)
| (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 = 1)
(ii) (1) (10 otimes m = 2)
By comparison, (10 otimes 9 = 2).
(2) (n otimes n = 4)
From the table, (9 otimes 9 = 4)
Hence, n = 9.
(b) Let the fraction = (frac{m}{n})
(frac{m}{n} = frac{3}{4}….. (1))
(implies n = frac{4m}{3} ……. (2))
(2m = n + 34 …… (3))
Put (2) in (3),
(2m = frac{4m}{3} + 34)
(2m – frac{4m}{3} = 34 implies frac{2m}{3} = 34)
(m = frac{34 times 3}{2} = 51)
(n = frac{4m}{3} = frac{4 times 51}{3} = 68)
Hence, the fraction = (frac{51}{68}).