Remainders

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There are two nunbers a and b both are divided by another number 10 and we get remainder *1 and 9 respectively . Now what will be the remainder of a b 10 \frac{ab}{10}

9 10 11 8

Aman Real by Aman Real , 22, India

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2 solutions

a=1mod10... (1)

b=9mod10... (2)

multiplying (1) and (2) together gives

ab = 9mod10

So remainder of ab/10 is 9.

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Akiva Weinberger
May 4, 2015

a = 10 k 1 + 1 a=10k_1+1 for some k 1 k_1 .

b = 10 k 2 + 9 b=10k_2+9 for some k 2 k_2 .

a b 10 = ( 10 k 1 + 1 ) ( 10 k 2 + 9 ) 10 \dfrac{ab}{10}=\dfrac{(10k_1+1)(10k_2+9)}{10}

a b 10 = 100 k 1 k 2 + 90 k 1 + 10 k 2 + 9 10 \phantom{\dfrac{ab}{10}}=\dfrac{100k_1k_2+90k_1+10k_2+9}{10}

a b 10 = 10 k 1 k 2 + 9 k 1 + k 2 + 9 10 \phantom{\dfrac{ab}{10}}=10k_1k_2+9k_1+k_2+\dfrac9{10} so the remainder is 9 9 .

(Or, if you know modular arithmetic, you get that result much faster.)

1 Helpful 0 Interesting 0 Brilliant 0 Confused

BTW i knew that the remainder of ab/10 and r1 r2/10 will be same so i just did 1 9/10

Aman Real - 6 years, 1 month ago

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