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PAYAL

· started a discussion

· 1 Months ago

Expalin properly (Question: 110042.0 )

please provide detailed solution

Question:

The remainder when 2851 × (2862)2 × (2873)3 is divided by 23 is:

Options:
A) 18
B) 19
C) 20
D) 21
E) 22
Solution:
Ans: (a) \(\cfrac{2851\times(2862)^2\times(2873)^2}{23}\ \ \ \ \ \Rightarrow \cfrac{22 \times10\times10\times21\times21\times21}{23}\)

\(\cfrac{22\times8\times441\times21}{23}\Rightarrow \cfrac{22 \times21\times8\times4}{23}\)

\(\cfrac{462 \times32}{23} \Rightarrow \cfrac{2\times9}{23}\)

Remainder is 18.

Knowledge Expert

· commented

· 1 Months ago

Dear Student,

You may solve it quickly by applying modulus or modulo operation Method :
When we say, divide 3 by 2, in modulus you can write that as,
3mod2.
Now, the remainder will be 3, 1, -1,....and so on. Please, note that i only wrote the possibilities here and only remainders. Now, simple rules in modulus-
1. (AxB)modC = (AmodC x BmodC)modC
2. A^BmodC = (AmodC)^BmodC
We shall be using these rules to solve your sum and find the remainder. So, if we write the sum using modulo, we have,
(2851 x 2862^2 x 2873^3)mod23
= [2851mod23 x 2862^2mod23 x 2873^3mod23]mod23
=[2851mod23 x (2862mod23)^2mod23 x (2873mod23)^3mod23]mod23
= [(-1) x (10)^2mod23 x (-2)^3mod23]mod23
=[(-1) x 100mod23 x (-8)mod23]mod23
=[(-1) x 8 x (-8)]mod23
= 64mod23
So, the remainder is, 18 and other possibilities too. Actually, there may be infinite number of remainders for any division.

Hope you have a great learning with us!!

All the Best,
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