What is the value of : -11 modulo 10

-11 mod 10

Thanks Heureka,

I do not know if I will ever get the hang of doing these formally but this is an easy one that you can do in your head.

This is mod 10 so think of a circle like a clock only instead of the numbers going to 12 they only go to 10 and then they start again.  Also 10 is really 0.

-11 = -10-1

Start at 0, go 10 places in a negative direction and that gets you back to 0 again.

Now go 1 more place in a negative direction and it puts you one number BEFORE 0.

That number is 9

so -11 mod 10 = 9

Some people call Mod maths, Clock maths !!

-11 modulo 10 = ¿?

\(\boxed{~ \begin{array}{rcll} a \pmod b &=& a - b \cdot \left[ \frac{a}{b} \right] \qquad [ \dots ] = \text{ integer part} \end{array} ~}\)

\(\begin{array}{rcll} a \pmod b &=& a - b \cdot \left[ \frac{a}{b} \right] \qquad & | \qquad a = (-11) \qquad b = 10 \\\\ (-11) \pmod {10} &=& (-11) - 10 \cdot \left[ \frac{-11}{10} \right] \\ &\equiv& (-11) - 10 \cdot [ -1.1 ] \\ &\equiv& (-11) - 10 \cdot ( -1) \\ &\equiv& (-11) + 10 \\ &\equiv& -1 \\ &\equiv& -1 + 10 \\ &\equiv& 9 \\ (-11) \pmod {10} &=& 9 \end{array}\)