How to calculate $1573 \pmod 5$?

Question

How to calculate $1573 \pmod 5$

I don't understand how to calculate modulus using a calculator. Is there any way I can rewrite this expression in multiple different ways to maybe see it easier?

Answer 1

In general, here is an algorithm you can use to calculate any number $N\pmod p$ on a dollar-store calculator. Let's take $N=1573$ and $p=7$ just for an example:

1. Enter $N \div p$. In our example, $1573 \div 7 = 224.714286$.
2. Remove the fractional part by rounding down to the next smallest integer. For a positive number, just clip off the decimal; for a negative number, rounding down means going up to the next higher ordinal - e.g., $-3.2$ rounds to $-4$. For this example, I cleared and entered $224$.
3. Now multiply by $p$. In our example, $224 \times 7 = 1568$. This value is the integer closest to $N$ that is also divisible by $p$.
4. Subtract $N$ from this, and the negation of this is the modulo value. $1568-1573 = -5$, therefore $1573 \equiv 5\pmod 7$.

Answer 2

You can use the fact that modulo is additive. I mean

$$1573 \text{ mod } 5 = 1570 \text{ mod } 5 + 3 \text{ mod } 5$$

Answer 3

Note that $$ \frac{1573}{5} = \frac{1570}{5} + \frac35 $$ and we are specifically after the remainder of this division.

Answer 4

Integral division of 1573 by 5 gives 314 with remainder 3. Thus $1573 = 314\cdot 5 + 3$ and so $1573\equiv 3\mod5$. But it can be done easier by just looking at the last digit of 1573.