Question

Let S = {1, 2, 3, ..., 10}, then Match the following:The number of subsets {x, y} of S so that x
^3
+ y
^3 is divisible by 3.​

Answer 1

Given :  S = { 1 , 2 , 3....................., 10 }

To find: The number of subsets {x, y} of S so that x so that x³ + y³  is divisible by 3

Solution:

1³   = 3k + 1

2³   = 3k - 1

3³ = 3k

4³ = 3k + 1

5³ = 3k - 1

6³ = 3k

7³ = 3k + 1

8³ = 3k-1

9³ = 3k

10³ = 3k + 1

x³ + y³  is divisible by 3

3k + 1   + 3k - 1  is divisible by 3

3k + 1   can be selected in 4 ways

3k - 1 ca be selected in 3 ways

4 * 3 = 12 Ways

(1 , 2)  . ( 1, 5) , ( 1, 8 )

(4 , 2) , (4 , 5) , ( 4, 8)

(7 , 2 ) , ( 7, 5) , ( 7 , 8)

(10 , 2) , (10 , 5) , ( 10 , 8)

3k + 3k is divisible by 3

³C₂ = 3 ways

(3 , 6) , ( 3, 9) , ( 6, 9)  - 3 ways

12 + 3 = 15  subsets

if order is also reversed  then total 30 subsets

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