Question

Write the cube of 7 natural number which are of the form 3n+ 1(i.e, 4,7,10,13...) and verify that "the cube of a natural number of the form 3n + 1 when divided by 3 leaves remainder 1

Answer 1

Step-by-step explanation:

write the cube of 7 natural number which are of the form 3n+1 and verify. that the cube of a natural number of the form 3n+1 when divided by 3 leaves remainder 1

Number    Cube    Quotient      Remainder

1                 1              0                 1

4                64           21                1

7               343          114                1

10           1000          333               1

13           2197          732                1

16           4096         1365               1

19           6859         2286               1

(3n + 1)³ =  (3n)³ + 1³ + 3(3n)(1)(3n + 1)

= 3 * 9 * n³  + 3 * 3n(3n+ 1)  + 1

= 3n * 3 ( 3n² + 3n + 1)  + 1

First term has 3 has factor so divisible by 3

second term = 1

so remainder will be 3  when divided by 3