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)

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

Answer 2

HEY DEAR ... ✌️

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=) Given, A is a single digit number .

If we assume , A = 1 then, A^3 = 1^3= 1 ,
Here , we see unit digit of A and A³ are same.

If we assume A = 2 then, A^3 = 2^3 = 8 , Here , we see unit digits are different .

If we assume A = 3 => A^3 = 27 ( wrong )

If we assume A = 4 => A^3 = 64 ( correct )

If we assume A = 5 => A^3 = 125 ( correct )

If we assume A = 6 => A^3 = 216 ( correct )

If we assume A = 7 => A^3 = 343 ( wrong )

If we assume A = 8 => A^3 = 512 ( wrong )

If we assume A = 9 => A^3 = 729( correct )

Hence, there are five solutions where unit digits of A and A^3 are same.

Now , x = 5

again, if we take A = 1 => A^2 = 1 and A^3 = 1 ( correct )

If we take A = 2 => A^2 = 4 and A^3 = 8 ( wrong )

If we take A = 3 => A^2 = 9 and A^3 = 27 ( wrong )

If we take A = 4 => A^2 = 16 and A^3 = 64 ( wrong )

if we take A = 5 => A^2 = 25 and A^3= 125( correct )

If we take A = 6 => A^2 = 36 and A^3 = 216 ( correct )

If we take A = 7 => A^2 = 49 and A^3 = 343 ( wrong )

If we take A = 8 => A^2 = 64 and A^3 = 512 ( wrong )

If we take A = 9 => A^2 = 81 and A^3 = 729 ( wrong )

Here , we see , there are three solutions where unit digits A^2 and A^3 are same .
So , y = 3

Hence , x = 5 and y = 3
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HOPE , IT HELPS ... ✌️