Question

If x and y are the number of possibilities that A can
assume such that the unit digit of A and A3 are
same and the unit digit of A2 and A3 are same
respectively, then the value of x – y is (where A is
a single digit number)
(1) 4 (2) 2
(3) 3 (4) 5

Answer 1

Given, A is a single digit number.

if A = 1 then, A³ = 1³ = 1 , here you can see unit digit of A and A³ are same.

if we take A = 2 then, A³ = 2³ = 8 , unit digits are not same.

if we take A = 3 => A³ = 27 ( × )

if we take A = 4 => A³ = 64 ( ✓ )

if we take A = 5 => A³ = 125 (✓)

if we take A = 6 => A³ = 216 (✓)

if we take A = 7 => A³ = 343 ( × )

if we take A = 8 => A³ = 512 ( × )

if we take A = 9 => A³ = 729( ✓)

hence, there are five possible solution where unit digits of A and A³ are same.

so, x = 5

again, if we take A = 1 => A² = 1 and A³ = 1 (✓)

if we take A = 2 => A² = 4 and A³ = 8 ( × )

if we take A = 3 => A² = 9 and A³ = 27 ( × )

if we take A = 4 => A² = 16 and A³ = 64 ( × )

if we take A = 5 => A² = 25 and A³ = 125( ✓ )

if we take A = 6 => A² = 36 and A³ = 216 ( ✓)

if we take A = 7 => A² = 49 and A³ = 343 ( × )

if we take A = 8 => A² = 64 and A³ = 512 ( × )

if we take A = 9 => A² = 81 and A³ = 729 ( × )

here we can see , there are three possible solution where unit digits of A² and A³ are same. so, y = 3

hence, x - y = 5 - 3 = 2

So, option (2) is correct.

Answer 2

HEY DEAR ... ✌️

___________________________

Thanks For Your Question ...

==============================

Given in the question ,

A is a single digit number.

=) If A = 1 then, A³ = 1³ = 1 , here you can see unit digit of A and A³ are same.

By assuming ,

=) If we take A = 2 . Then, A³ = 2³ = 8 , unit digits are not same.

Now ,

If we take A = 3 => A³ = 27 ( wrong )

If we take A = 4 => A³ = 64 ( correct )

If we take A = 5 => A³ = 125 (correct )

If we take A = 6 => A³ = 216 (correct )

If we take A = 7 => A³ = 343 ( wrong )

If we take A = 8 => A³ = 512 ( wrong )

If we take A = 9 => A³ = 729( correct )


so, value of x = 5

Therefore again,

if we take A = 1 => A² = 1 and A³ = 1 (correct )

if we take A = 2 => A² = 4 and A³ = 8 ( wrong )

if we take A = 3 => A² = 9 and A³ = 27 ( wrong )

if we take A = 4 => A² = 16 and A³ = 64 ( wrong )

if we take A = 5 => A² = 25 and A³ = 125( wrong)

if we take A = 6 => A² = 36 and A³ = 216 ( correct )

if we take A = 7 => A² = 49 and A³ = 343 ( wrong )

if we take A = 8 => A² = 64 and A³ = 512 ( wrong )

if we take A = 9 => A² = 81 and A³ = 729 ( wrong )

so, value of y = 3

So , x - y
5-3
2

___________________________

HOPE , IT HELPS ... ✌️