\(5^{16} - 3^{16}\)

\(5^8*2 - 3^8*2\)

\((5^8 +3^8) (5^8 - 3^8)\)

now just break down 5^8 - 3^8.

apply \(a^2 -b^2.\)

at least we got : (\(5^8 +3^8) (5^4 +3^4) (5^2 +3^2) ( 5 +3 ) (5-3)\)

Now divide :

\((5^8 +3^8) (5^4 +3^4) (5^2 +3^2) ( 5 +3 ) (5-3) /16.\)

\((5^8 +3^8) (5^4 +3^4) (5^2 +3^2) *8*2 /16.\)

Reminder is 0.

A is the correct answer.

Solution

To find:
We are asked to find out,

• The remainder when \(5^{16} – 3^{16}\) is divided by 8

Approach and Working:

• \(5^{16} – 3^{16}\) can be written as \((5^8)^2 – (3^8)^2\)

We know that, \(a^2 – b^2 = (a + b) * (a - b)\)

• Thus, \((5^8)^2 – (3^8)^2 = (5^8 + 3^8) * (5^8 – 3^8) = (5^8 + 3^8) * (5^4 + 3^4) * (5^4 – 3^4) = (5^8 + 3^8) * (5^4 + 3^4) * (5^2 + 3^2) * (5^2 - 3^2) = (5^8 + 3^8) * (5^4 + 3^4) * (5^2 + 3^2) * (5 + 3) * (5 – 3)\)
• So, \(5^{16} – 3^{16} = 16 * (5^8 + 3^8) * (5^4 + 3^4) * (5^2 + 3^2) * (5^2 - 3^2)\)

If you observe carefully, the above expression is a multiple of 8

• Therefore, the remainder will be 0

Hence the correct answer is Option A.

Answer: A


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