Integer Division/Examples/Division by -7

Examples of Integer Division

$\ds 1 \div \paren {-7}$

$=$

$\ds 0$

$\ds \rem 1$

$\ds -2 \div \paren {-7}$

$=$

$\ds 1$

$\ds \rem 5$

$\ds 61 \div \paren {-7}$

$=$

$\ds -8$

$\ds \rem 5$

$\ds -59 \div \paren {-7}$

$=$

$\ds 9$

$\ds \rem 4$

We have that:

$29 = 3 \times 8 + 5$

$\ds 1$

$=$

$\ds 0 \times \paren {-7} + 1$

$\ds -2$

$=$

$\ds 1 \times \paren {-7} + 5$

$\ds 61$

$=$

$\ds \paren {-8} \times \paren {-7} + 5$

$\ds -59$

$=$

$\ds 9 \times \paren {-7} + 4$

Hence the result by definition of integer division.

$\blacksquare$

1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.1$ The Division Algorithm