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Any help here? i have an upcoming exam, and the question in some of the exercises that im practicing on are (603 · 6004 + 60005) mod 6 is equal

I just dont understand how to do it. The way i saw it was to do [603 (MOD 6) ] * [6004 (MOD 6) ] There by using the formular n= qm + R 603 = 100*6 + 3 6004 = 1000*6+ 4 Then using these number 3*4 (MOD 6) 12 (MOD 6) 12 = 1*6 + 6 12 = 6 MOD 6 But the solutions says 5 ? I dont get it?

ABCCode
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2 Answers2

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${\rm mod}\ 6\!:\,\ 6x\!+\!\color{#c00}y\equiv \color{#c00}y,\ $ so $\ (6i\!+\!\color{#c00}a)(6j\!+\!\color{#c00}b)+6k\!+\!\color{#c00}c \,\equiv\, \color{#c00}{ab}+\color{#c00}c\ $ by Basic Congruence Rules.

Community
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Bill Dubuque
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$603=600+3 \equiv 0 +3 \equiv 3$ mod 6. $6004=6000+4 \equiv 0+4$ mod 6. $60005=60000+5 \equiv 0 +5 $ mod 6. Hence, $603 \times 6004+60005 \equiv 3 \times 4+5 \equiv 12+5 \equiv 5$ mod 6

mich95
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  • So my mistake was that i forgot to do the same with 60005? But was everything besides that correct? – ABCCode Jun 08 '15 at 01:24
  • yes @ABCCode... – janmarqz Jun 08 '15 at 01:27
  • There is another exercise which says exactly the same but with mod 10 (603 · 6004 + 60005) mod 10 is equal to? I did everything just as before and got the same 3 x 4 + 5 however it says the result is 7 in this case? – ABCCode Jun 08 '15 at 01:35
  • Then the result is $10 \times 5 \equiv 20 \equiv 18+2 \equiv 2$ mod 6 – mich95 Jun 08 '15 at 01:40
  • I still dont get it 10 x 5 why are you doing that? Im new to the topic and also understanding ≡ so if you break it into pieces it would be a lot easier to understand – ABCCode Jun 08 '15 at 01:46
  • $ab$ mod 6 is the same as $a$ mod $6 \times b$ mod $6$. Now, in our case, we've proven that $(603 \times 604 +60005) \equiv 5$ mod 6, so $10(603 \times 6004+60005)$ mod $6$ is $10$ mod $6 \times 5$ mod $6$ and hence is $4 \times 5$ mod 6, $20$ mod 6 hence 2, – mich95 Jun 08 '15 at 01:51
  • Are you sure that you got the questions?

    (603 · 6004 + 60005) mod 10

    I just can't seem to grasp it?

     – ABCCode Jun 08 '15 at 10:16
  • Because i wrote mod 10 (603 · 6004 + 60005) mod 10 so maybe you misunderstood – ABCCode Jun 08 '15 at 10:18
  • Ohhh! $603 \times 6004+60005 \equiv 10\times 60 +3 + 10 \times 600 +4+10 \times 6000 +5 \equiv 3+4+5 \equiv 12 \equiv 2$ mod $10$. – mich95 Jun 08 '15 at 11:47