Linear congruences (single)

6

1. Determine all values of \(x\) for which \(16 x \equiv 5 ( \bmod 101 )\).
2. Solve
   1. \(95 x \equiv 6 ( \bmod 101 )\),
   2. \(95 x \equiv 5 ( \bmod 101 )\).

1

1. Evaluate \(13 \times 19\) modulo 31 .
2. Solve the linear congruence \(13 x \equiv 9 ( \bmod 31 )\).

1. Tiles are sold in boxes with 21 tiles in each box.

The tiles are laid out in \(x\) rows of 5 tiles and \(y\) rows of 6 tiles.
All the tiles from a box are used before the next box is opened.
When all the rows of tiles have been laid, there are \(n\) tiles left in the last opened box.

1. Write down a congruence expression for \(n\) in the form $$a x + b y ( \bmod c )$$ where \(a\), \(b\) and \(c\) are integers. Given that
   • exactly 43 rows of tiles are laid
   • there are no tiles left in the last opened box
   • use your congruence expression to determine the minimum number of rows of 6 tiles laid.

1 Solve \(12 x \equiv 3 ( \bmod 99 )\).