​$$97^{83} - 1 \text{ when divided by } 96 \text{ would leave a reminder}$$​​

Correct option is D

Given:

Expression: $$97^{83} - 1$$​​

Divisor: 96

Concept Used:

Remainder rule using binomial form

Property: $$(a+1)^n$$​ when divided by (a) leaves remainder (1)

Formula Used:

If (x = a+1), then
​$$(a+1)^n \equiv 1 \pmod a$$​​

Solution:

97 = 96 + 1

​$$97^{83} = (96+1)^{83}$$​​

When $$(96+1)^{83}$$​ is divided by (96), the remainder is:

$$1^{83}$$​ = 1

Now subtract the constant:

​$$97^{83} - 1 \equiv 1 - 1 \equiv 0 \pmod{96}$$​​

Remainder = 0

Alternate Solution (Exam Trick Method):

Any number of the form (a+1)ⁿ - 1 is always divisible by (a).

Here, a = 96

$$(96+1)^{83}$$​ - 1 ➔ exactly divisible by 96

Remainder = 0