Correct Answer - Option 2 : 5
Concept:
If a number x is divided by y and leaves remainder z, then it can be written as x = yp + z; where p is any natural number
Calculation:
Since a and b gives remainder 4 and 5 on division by 6 respectively
Hence, a = 6m + 4; b = 6n + 5, where m and n are any natural numbers
a2 + b2 = (6m + 4)2 + (6n + 5)2
⇒ a2 + b2 = 36m2 + 16 + 48m + 36n2 + 25 + 60n
⇒ a2 + b2 = 36(m2 + n2) + 41 + 48m + 60n
Since, 36 is divisible by 6, hence 36(m2 + n2) is also divisible by 6
Similarly, 48m and 60n are also divisible by 6
41 = 36 + 5, hence it will leave 5 as a remainder
∴ a2 + b2, when divided by 6, will leave remainder 5