Question

243 x 245 x 247 x 249 x 251 divided by 12 and find the remainder only.
can you show how the steps goes too, it's using the Remainder theorem.​

Answer 1

✿$\underline{\textbf{\textsf{ \purple{Solution}:- }}}$

We have to find the remainder of,

$\sf{\dfrac{243 * 245 * 247 * 249 * 251}{12}}$

The best approach to solve this question is by using the concept of Negative Remainder.

$\sf{\\}$

$\underline{\texttt{ \red{Negative Remainder} :- }}$

Let's consider an example,

$\sf{\dfrac{15}{4}}$

$\quad \quad \bullet$If we consider $\sf{ 4 * 3 = 12 }$, and hence remainder will be $\sf{15 - 12 = \underline{3}}$, then 3 is positive remainder.

$\quad \quad \bullet$If we consider $\sf{ 4 * 4 = 16 }$, and hence remainder will be $\sf{15 - 16 = \underline{-1}}$, then -1 is negative remainder.

The benefit of negative remainder is that, when positive remainder is large, we can use the negative remainder if it's magnitude is small. As you can see, between -1 and +3, for calculations such as multiplication, it is easier to work with -1 than with +3.

$\star$There is a shortcut for finding the negative remainder. Suppose you have got the positive remainder here +3. Then for saving time, just subtract positive remainder from denominator and then put a negative sign. For example, here, positive remainder is +3. Then do (4 - 3) = 1 and then put a negative sign. So the final negative remainder will be -1.

$\sf{\\}$

Coming to the question,

$\sf{\dfrac{243 * 245 * 247 * 249 * 251}{12}}$

$\sf{12 * 20 = 240}$, and $\sf{12 * 21 = 252}$

So, we can use negative remainder for 247, 249 and 251

• For 243, $\sf{12 * 20 = 240, \:hence\:243 - 240 = \blue{+3}}$
• For 245, $\sf{12 * 20 = 240, \:hence\:245 - 240 = \blue{+5}}$
• For 247, $\sf{12 * 21 = 252, \:hence\:247 - 252 = \blue{-5}}$
• For 249, $\sf{12 * 21 = 252, \:hence\:249 - 252 = \blue{-3}}$
• For 251, $\sf{12 * 21 = 252, \:hence\:251 - 252 = \blue{-1}}$

Now, it can be written as,

$\sf{\dfrac{243 * 245 * 247 * 249 * 251}{12}}$

$\longrightarrow \sf{\dfrac{ (+3) * (+5) * (-5) * (-3) * (-1) }{12}}$

$\longrightarrow \sf{\dfrac{-225}{12}}$

We know that,

$\sf{12 * 19 = 228, \: so \: 225 - 228 = -3 (negative remainder)}$

$\longrightarrow \sf{- (-3)}$

Hence the remainder is,

$\longrightarrow \underline{\underline{\sf{ \green{Remainder\:=\:3} }}}$

Answer 2

Step-by-step explanation:

using remainder theorem it is pretty easy