The remainder when 40 + 41 + 42 + 43 + …….. + 440 is divided by 17 is:
a) 0
b) 16
c) 4
d) None of these
Answer: d
Explanation: Let S be the sum of the expression,
S = 40 + 41 + 42 + 43 + …….. + 440
S = (1 + 4 + 16 + 64) + 44 (1 = 4 + 16 + 64) + …… + 436 + 440
Since, (1 + 4 + 16 + 64) = 85, is divisible by 17. Hence, except 440 remaining expression is divisible by 17.
$$\frac{{{4^{40}}}}{{17}} \to \frac{{{{\left( {{4^4}} \right)}^{10}}}}{{17}} \to \frac{{{1^{10}}}}{{17}}$$
Remainder = 1
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