Question

If x and y are the number of possibilities that A can assume such that the unt of A and A3 are same and the unit of A and A3 are same respectively, then the value of x-y is (where A is a single digit number)

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