Show that 9453(6824)$\equiv$6782(5675341)$\equiv$2 (mod 5)

Question

I am very new to modular arithmetic and I am not entirely sure what this question is asking me to do, or how you would go about showing what it is looking for. I apologise if this is very simple, but I am looking for some clarification on what this actually means. Thanks.

Do you know that $a \equiv b$ (mod $p$) and $c \equiv d$ (mod $p$) implies $ac \equiv bd$ (mod $p$)? — Loafy Loafer, Nov 10 '19 at 09:07
Yeah, I think I have seen this before, but how do you apply it to this problem? — Jamminermit, Nov 10 '19 at 09:12

score 1 · Answer 1 · answered Nov 10 '19 at 09:16

1

Hint: $a\equiv r$ (mod $p$), where $r$ is the remainder when $a$ is divided by $p$.

What are the remainders when $9453, 6824, 6782$ and $5675341$ are divided by $5$?

answered Nov 10 '19 at 09:16

Loafy Loafer

937
8
25

So the remainders are 3, 4, 2 and 1 – Jamminermit Nov 10 '19 at 09:24
So can I say that 9453$\equiv$3 (mod 5) and 6824$\equiv$4 (mod 5) and 6782 $\equiv$2 (mod 5) and 5675341 $\equiv$1 (mod 5)? And using the rule above, then 3(4)$\equiv$2(1) (mod 5) so 12$\equiv$2 (mod 5)? – Jamminermit Nov 10 '19 at 09:31
@Jamminermit that is perfectly correct. – YiFan Tey Nov 10 '19 at 09:45
which is the same as saying $12-2$ is divisible by 5. – Nov 10 '19 at 11:57
@Jamminermit Another way is to compute the decimal units digits then reduce that $\bmod 5,,$ i.e. $, n\bmod 5 = (n\bmod 10)\bmod 5,,$ a special case of the method of simpler multiples. – Bill Dubuque Nov 10 '19 at 23:35

Allawonder · Answer 2 · 2019-11-11T07:20:50.650

1

Hint. It is relatively easy to reduce numbers in decimal form modulo $5.$ Just expand out and note that any positive power of $10$ vanishes modulo $5.$ Thus, the residues are just the residues of the units digits, which are easy to do. Can you now proceed?

edited Nov 11 '19 at 07:20

answered Nov 10 '19 at 10:42

Allawonder

13,327

The residue $\bmod 5,$ is the units digits reduced $\bmod 5,,$ i.e. $, n\bmod 5 = (n\bmod 10)\bmod 5,,$ see my comment in Jerry's answer. – Bill Dubuque Nov 11 '19 at 02:23
@BillDubuque Thanks. This is what I should have said. – Allawonder Nov 11 '19 at 07:20

Show that 9453(6824)$\equiv$6782(5675341)$\equiv$2 (mod 5)

2 Answers2