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$60-88 \equiv \,\,? \pmod 5$

$60-88 = -28$

Then what do I do?

Please tell me how to answer this question. Thanks.

TMM
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MethodManX
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2 Answers2

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When you have a negative number, like $-28 \pmod 5$, all you really need to do keep adding the modulus, or an integer multiple of the modulus $m$, to the negative number until it is zero or in $\mathbb Z_m = \{0, 1, \cdots, m-1\}.\;$

In this case, we can add any multiple of the modulus $5$ to $-28$ until we obtain zero or a positive number, if you are looking to represent the equivalence class of $-28$ modulo $5$ with the least positive integer $x$, $\;x \in \{0, 1, 2, 3, 4\}$.

So given $$60 - 88 = -28 \equiv x\pmod 5\tag{1}$$ and wanting to find such a representative $x$, we know that $5k \equiv 0 \pmod 5$, and simply choose $k = 6$ as our multiple of $5$ so we have $$5 \cdot 6 = 30 \equiv 0 \pmod 5\tag{2}$$

Adding $(1), (2)$:

$$30 - 28 = 2 \equiv x \pmod 5$$

That is, $60 - 88 = - 28 \equiv 2 \pmod 5$.

What this all means is that any multiple $n$ of 5, added to $2$ is congruent modulo $5$ to $2$, including $-28$: The solutionS to all $$x \equiv 2 \pmod 5 = \{x\mid x = 2 \pm 5n, n \in \mathbb Z\}$$

amWhy
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  • @ amWhy so the answer is 2 ? – MethodManX May 27 '13 at 19:07
  • Yes, exactly. The answer is $2$, but every multiple of 5 added to or subtracted from $2$ is also congruent to $2$, mod 5. but I suspect you are needing $2$, which is the least non-negative solution to your problem, mod 5. – amWhy May 27 '13 at 19:09
  • @amWhy: Always providing helpful comments and descriptions ... +1 – Amzoti May 28 '13 at 00:35
  • @amWhy

    50-99 = (mod 5)

    = -49 + 50

    so answer is 1?

     – MethodManX May 29 '13 at 17:31
  • Yes, you are correct. Nice work. Please feel free to accept helpful answers: just click on the $\large \checkmark$ to the left of the answer you'd like to accept. – amWhy May 29 '13 at 18:09
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As Key Ideas pointed in the comments...

$60-88 \equiv \,\,x \pmod 5$

$-28 \equiv \,\,x \pmod 5$

$-28+30 \equiv \,\,x+30 \pmod 5$

$2 \equiv \,\,x \pmod 5$

$x=5n-2$ with $n\in\mathbb{Z}$

Maazul
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