Aptitude - Numbers - Discussion

22 + 2 divided by 3.
= 4 + 2,
= 6.
so when 6 divided by 3 then remainder = 0.

So by doing calculations like this we can get:

22 + 2 divided by 3 then remainder = 0.
33 + 3 divided by 4 then remainder = 2.
44 + 4 divided by 5 then remainder = 0.
55 + 5 divided by 6 then remainder = 4.

So by observing above examples we can say;

xx + x is divided by (x+1) then the remainder is (x-1) where x is odd number.

So, now we can say when 6767 + 67 is divided by 68 then remainder is 66.

(67^67+67)/68 = 67^67/68 + 67/68.
= (67+67)/68.

a^n/(a+1) = a when n is odd.
= 66(ans),
= 1 when n is even.

Is there any Short trick?

a^b is a raise to the power b.

To understand this question we must take a simpler example.
for ex- (3x3)/4

3 = (4-1),
Therefore the question becomes
(4-1)(4-1)/4.
On multiplying (4^2-4-4+1)/4
Note that all the terms in the expansion are completely divisible by 4 except 1.hence Remainder will be 1.

Now take expression for example- 3x3x3 /4.
again (4-1)(4-1)(4-1)/4,
(4^2-4-4+1)(4-1) / 4,
= 4x(4^2-4-4+1)-1x(4^2-4-4+1),
= 4x(4^2-4-4+1) + (-4^2)+4+4-1.

Here the first term is divisible by 4 and in the second term after multiplication by -1 only +1 remains which is not divisible.

Carrying out this process only 1 will remain as the remainder and its sign will depend on the power of the numerator. when the power of 3 was 2, the remainder was +1 and when the power was 3, the remainder was -1.

This means when the power of numerator is odd, -1 will remain and when the power will be even, +1 will remain.

Using this in our example:- (67^67)/68.

-1 will remain because the power of the numerator is odd.
and we also have a +67 in the numerator. therefore, total remaining in the numerator is -1+67 = 66. Which is the remainder and our answer.

Thank you @Anuj.

It's very easy, Thank you @Mahi.

Give more clarity about this question.

Why do we subtract 2 from 68?

The formula is a^n + b^n is divisible by a + b if n is odd.

Why do we subtract 2 the remainder from 68, can anyone explain this?