Kindly provided by UKMT What is the remainder when the above total is divided by 8? (no calculators please!)
9 Comments
simon
14/2/2015 02:17:13 am
My immediate thought is 1. Basically you have 4 x odd / 8 which reduces to odd/2 which has remainder 1.
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Andrey Dyukmedzhiev
14/2/2015 02:33:43 am
Since 5 ≡ -3 (mod 8), 7 ≡ -1 (mod 8) and 3² = 9 ≡ 1 (mod 8) we have:
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simon
14/2/2015 03:08:51 am
Yes, 4 is correct. I should have multiplied my remaining 1/2 by 8.
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14/2/2015 05:16:28 am
Nice one, Paul!
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Dr Baljinder Singh Saini
15/2/2015 02:19:26 am
2^2/8=1/2 Therefore remaining 3^3×5^5×7^7=an odd number.
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Rudy
15/2/2015 03:44:06 am
2*2 = 4,
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Satwaki Chanda
15/2/2015 06:41:07 am
The product of the last three terms is an odd number so the answer is of the form 4(2n+1) = 8n + 4. So remainder 4.
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Paul
15/2/2015 09:44:33 pm
Thanks to Simon, Andrey, Anand, Baljinder, Rudy and Satwaki. It's great to see your different approaches (I always find modulo arithmetic fascinating). Let's see if anyone else has a new approach. Much obliged to you all as always. Paul
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Tom T
16/2/2015 01:30:52 pm
2^2 is 4. If I multiply 4 by an odd number, the result is 4 mod 8. All of 3, 5 , and 7 and all their powers are odd numbers, so none of them multiplying changes 4 mod 8 to anything else.
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