Question

19 to the power 200 if divided by 20 what is the reminder

Answer 1

Hi there!
Here's the answer:

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Remainder of [19^(200)] ÷ 20 :

For every expression, there comes attached a specific cyclicity of remainders.

To find cyclicity, we keep finding remainders until some remainder repeats itself.

19^(200) ÷ 7

No. ÷ 7 :
19__19²__19³__19^4__19^5__19^6__(19^7)

Remainder:
5___4____6____2____3_____1____(5)

19^7 gives same remainder as 19, when divided by 7 & this cycle continues.

•°• Cyclicity = 6.

So any power of 6 or a multiple of 6 will give a remainder 1.

Express 19^200 separately as 19^(6x+y)
i.e., 19^(6x) × 19^(y)

=> 6 × 33 = 198
=> 200 = 198 + 2

• 19^(200) ÷ 7 = [19^(198) ÷ 7 ] × [19² ÷7 ]

• Rem of{ 19^(200) ÷ 7 }
= Rem of{ 19^(198) ÷ 7 } × Rem of{ 19² ÷ 7 }

¶¶¶ This is according to remainder theorem,
>>> The product of any 2 or more than 2 no.s has the same remainder when divided by any Natural No., as the product of their remainders.

=> Rem of{ 19^(200) ÷ 7 }
= 1 × 4
= 4

•°• Final remainder = 4.

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Hope it helps