Question

5^2021 + 2(7^2021) + 9^2021 + 11^2021 + 5(13^2021) + 15^2021 divided by 8, then the remainder is?​

Answer 1

5²⁰²¹ + 2(7²⁰²¹) + 9²⁰²¹ + 11²⁰²¹ + 5(13²⁰²¹) + 15²⁰²¹ divided by 8 leave remainder 7

Given:

• 5²⁰²¹ + 2(7²⁰²¹) + 9²⁰²¹ + 11²⁰²¹ + 5(13²⁰²¹) + 15²⁰²¹
• Divided by 8

To Find:

• Remainder

Solution

Finding remainder for 5²⁰²¹ , 7²⁰²¹ , 9²⁰²¹ , 11²⁰²¹  , 13²⁰²¹ and 15²⁰²¹ when divided by 8

Step 1:

Rewrite 5²⁰²¹ as 5²⁰²⁰ * 5

= (5²)¹⁰¹⁰ * 5

= (25)¹⁰¹⁰ * 5

= (8 * 3 + 1)¹⁰¹⁰ * 5

(8 * 3 + 1)¹⁰¹⁰ Divided by 8 will leave remainder 1 as except 1¹⁰¹⁰ = 1 , all terms will have 8 as factor.

Hence remainder = 1 * 5 = 5  when 5²⁰²¹  Divided by 8

Step 2:

Rewrite 7²⁰²¹ as 7²⁰²⁰ * 7

= (7²)¹⁰¹⁰ * 7

= (49)¹⁰¹⁰ * 7

= (8 * 6 + 1)¹⁰¹⁰ * 7

(8 * 6 + 1)¹⁰¹⁰ Divided by 8 will leave remainder 1 as except 1¹⁰¹⁰ = 1 , all terms will have 8 as factor.

Hence remainder = 1 * 7 =7  when 7²⁰²¹  Divided by 8

Hence 2(7) = 14 = 8 * 1 + 6

=> 2(7²⁰²¹) divided by 8 will leave remainder 6

Step 3:

9²⁰²¹ divided by 8

= (8   + 1)²⁰²¹

Divided by 8 will leave remainder 1 as except 1¹⁰¹⁰ = 1 , all terms will have 8 as factor.

Hence remainder = 1 when 9²⁰²¹  Divided by 8

Step 4:

Rewrite 11²⁰²¹ as 11²⁰²⁰ * 11

= (11²)¹⁰¹⁰ * 11

= (121)¹⁰¹⁰ * 11

= (8 * 15 + 1)¹⁰¹⁰ * 11

(8 * 15 + 1)¹⁰¹⁰ Divided by 8 will leave remainder 1 as except 1¹⁰¹⁰ = 1 , all terms will have 8 as factor.

Hence remainder = 1 * 11 = 11 = 8 + 3 hence reminder 3  when 11²⁰²¹  Divided by 8

Step 5:

Rewrite 13²⁰²¹ as 13²⁰²⁰ * 13

= (13²)¹⁰¹⁰ * 13

= (169)¹⁰¹⁰ * 13

= (8 * 21 + 1)¹⁰¹⁰ * 13

(8 * 21 + 1)¹⁰¹⁰ Divided by 8 will leave remainder 1 as except 1¹⁰¹⁰ = 1 , all terms will have 8 as factor.

Hence remainder = 1 * 13 = 13 = 8 + 5 hence reminder 5  when (13)²⁰²¹  Divided by 8

5((13)²⁰²¹) remainder = 5(5) = 25 = 8 * 3 + 1  

Hence Remainder = 1  when 5((13)²⁰²¹)  divided by 8

Step 6:

Rewrite 15²⁰²¹ as 15²⁰²⁰ * 15

= (15²)¹⁰¹⁰ * 15

= (225)¹⁰¹⁰ * 15

= (8 * 28 + 1)¹⁰¹⁰ * 15

(8 * 28 + 1)¹⁰¹⁰ Divided by 8 will leave remainder 1 as except 1¹⁰¹⁰ = 1 , all terms will have 8 as factor.

Hence remainder = 1 * 15 = 15 = 8 + 7 hence reminder 7  when (15)²⁰²¹  Divided by 8

Step 7:

Add all remainders

5 + 6 + 1 + 3 + 1 + 7 = 23

23 = 8 * 2 + 7

Hence Remainder = 7

5²⁰²¹ + 2(7²⁰²¹) + 9²⁰²¹ + (11)²⁰²¹ + 5(13²⁰²¹) + (15)²⁰²¹ divided by 8 leave remainder 7

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