My take on solving this question :

Form an equation first. Treat 'A' as a digit.

96 x A = 76A

Now we know 'A' is in the ones place. So what we can do is to rewrite the number 76A by breaking it up into its constituents.

(Example : 546 = 540 + 6)

76A = 760 + A = 760 + 1 x A

So this tells us that 96 x A = 760 + 1 x A


96 'A's is equal to 760 plus 1 'A'

To get from 1 'A' to 96 'A's, we need to add 95 'A's

But realise that we also get to 96 'A's when we add 760.

This tells us that 760 equals 95 'A's

So 1 'A' = 760 ÷ 95 = 8


A is 8

Another way :


1.

When a digit is multiplied by 6, the only way to get the digit itself in the ones place of the product is :


① If the digit is 8
(Eg. 6 x 8 = 48)

② if the digit is 2
(Eg. 6 x 2 = 12)

③ if the digit is 4
(Eg. 6 x 4 = 24)

④ if the digit is 6
(Eg. 6 x 6 = 36)

⑤ if the digit is 0
(Eg. 6 x 0 = 0)

(Side note : notice that they are all even digits)

So A can only be 0,2,4,6 or 8.

Now,


We know that A cannot be 0 since the product of 96 and 0 will be 0 and not 960. So that leaves us with 2,4,6 or 8.


2.

For the number 96, the digit in the tens place is a '9'. We notice that the product of 96 and A is 76A, which has 76 as the digits in the hundreds and tens place respectively.

In order to get this '76' in the front, the '9' in 96 has to be multiplied by a digit which will give 7 as the first digit of the product.

And the only possibility is when A is 8, since 9 x 8 = 72

9 x 1 = 9
9 x 2 = 18
9 x 3 = 27
9 x 4 = 36
9 x 5 = 45
9 x 6 = 54
9 x 7 = 63
9 x 8 = 72 (the only possible case)
9 x 9 = 81
9 x 6 = 54


So A is 8.


(Familiarity with the times table will be good. Point 2 alone is actually sufficien to solve this question )

And yet another way :


96 x A

= 100 x A - 4 x A

= A00 - 4 x A

Since 96 x A = 76A,

A00 - 4 x A = 76A


Realise that 4 x A will never be higher than 36. (Highest Possible is 4 x 9 = 36)

So when 4 x A is subtracted from A00, the digit in the hundreds place of the result is only 1 lower than A.

Since 76A has 7 in the hundreds place, A can only be 8.

Check:

96 x 8

= 100 x 8 - 4 x 8

= 800 - 32

= 768

A: 1u

96 x A
= 96u

76A = 760 + 1u


96u = 760 + 1u
95u = 760
1u = 760/95 = 8

so A is 8.

K.I.S.S.

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