Solution of exercise on the topic of Remainder Theorem

1. Find the remainder of \(\frac{{18 + 17 – 15 + 130 + 138 – 178}}{{16}}\)
Sol:

3. Find the remainder of \(\frac{{71 + 80 + 89 – 107}}{9}\)
Sol:

4. Find the remainder of \(\frac{{341 \times 342 \times 343 \times 344}}{{13}}\)
Sol:

5. Find the remainder of \(\frac{{{{243}^{243}}}}{{244}}\)
Sol:

6. Find the remainder when 123456……..61 digits is divided by 16.
Sol:
To get remainder of division by 16 we need last 4 digits.
⟹ 1 ⎯ 9 ⟶ 9 digits
remaining digits = 61 – 9 = 52
starting from 10 further numbers have 2 digits
∴ \(\frac{{52}}{2}\) = 26 numbers are required of which first number is 10
∴ last number = 10 + (26 – 1).1 = 35
∴ last 4 digits are 3435
Remainder = \(\frac{{3435}}{{16}}\) = 11    Answer

7. Find the remainder when 12345……… till 31 digits is divided by 8.
Sol:
To know the remainder of division by 8 we only need last 3 digits.
⟶ 1 to 9 ⟹ 9 digits
Remaining digits = 31 – 9 = 22
each number after 9 will have 2 digits
∴ number required = \(\frac{{22}}{2}\) = 11
∴ last number = 10 + (11 – 1).1 = 20
∴ last 3 digits will be 920
Now check Remainder ⟹ 
\(\frac{{920}}{8}\) = 0     Answer

8. Find the remainder of \(\frac{{{8^{77}}}}{{17}}\)
Sol:
Method(1):
\(\frac{{{8^{77}}}}{{17}} = \frac{{{2^{231}}}}{{17}} = \frac{{{{\left( {{2^4}} \right)}^{57}}{{.2}^3}}}{{17}}\)

Method(2):
\(\frac{{{8^{77}}}}{{17}}\)
Φ(17) = 16 & 8 and 17 are co-prime
∴ \(\frac{{{8^{16}}}}{{17}}\) = 1 ⟵ Remainder

9. Find the remainder when 1! + 2! + 3! + ……….. + 100! is divided by
(i) 5          (ii) 6        (iii) 12
Sol:

10. Find the remainder when 84 × 86 × 399 × 432 is divided by 380.
Sol: