EgmatQuantExpert wrote:
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: AHi
egmat,
Can this problem be solved without using the identity.
(5^16 - 3^16) / 8 = ( 5^16/8 ) - ( 3^16/8 )
Now we make use of cyclic pattern and compute remainder.
5/8 = remainder is always 5
3/8 = remainder is 1 as pattern repeats after 2
So remainder is = 5-1=4
This will be divisible by 8 as 4 ^16 hence the remainder is 0
Please comment on the approach whether it is correct or not