Question

Number of natural numbers are such that a^3 – a^2 is a square of a natural number is

Answer 1

Answer:

ANSWERANSWER

Step-by-step explanation:

a and b are chosen from natural numbers.

N={1,2,3,4,.......}

Let p be a natural number.

then p is of the form 5k or

5k+1

5k−1

or

5k+2

5k−2

p

2

=

⎩

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎧

25K

2

25K

2

±10K+1

25K

2

±20K+4

So, The remainder of

5

P

2

would be either 0 or 1 or −1.

Case (i): Both the numbers a & b come from the set whose square gives reminder 0 with 5. Number of possibilities =

5

1

×

5

1

=

25

1

Case (i|): One no. 'a' gives reminder '1' with 5 and 'b' gives '−1' and vice versa.

So, Number of ways =

2

C

1

.(

5

2

)(

5

2

)

=

25

8

.

∴ Required probability =

25

9