Question
Find the remainder when 32025 is divided by 35.
Solution
We can use Euler’s theorem or cyclicity. φ(35) = φ(5 × 7) = 4 × 6 = 24 So 324 ≡ 1 (mod 35), and powers repeat every 24. Compute exponent modulo 24: 2025 ÷ 24 = 84 × 24 = 2016, remainder 9 So 32025 ≡ 39 (mod 35) Now compute 39 mod 35: 3¹ = 3 3² = 9 3³ = 27 3⁴ = 81 ≡ 81 − 70 = 11 3⁵ = 11×3 = 33 3⁶ = 33×3 = 99 ≡ 99 − 70 = 29 3⁷ = 29×3 = 87 ≡ 87 − 70 = 17 3⁸ = 17×3 = 51 ≡ 51 − 35 = 16 3⁹ = 16×3 = 48 ≡ 48 − 35 = 13 Remainder = 13.
More No. System Questions
- The sum of a positive number 'A' and its square is 156. Find the value of (A2 - 7A).
- The sum of three numbers is 114. On dividing the largest number by the smallest number, the result is 8, which is 1/3rd of the second largest number. Find ...
- Find the value of 'a' in the given data if the mean of the given data is 16. 10, 12, 18, 20, 15, 17, a, 14, 13, 19
- Sonal had a total income of Rs. 45000. She spent Rs. 5000 less than the amount she saved. What portion of her income did she spend in percentage?
- 28% of a certain number is added to 82% of the same number, resulting in 125% of 484. If 150 is subtracted from this number, what is 35% of the resulting v...
- Find the least number greater than 100 which leaves a remainder of 4 when divided by 5, 6, and 7.
- If '81x4y' is a five-digit number which is divisible by 72, then find the value of (3x + 5y).
- What is the highest possible sum of (x + y) if the number '7x542y6' is divisible by 12?
- If '57x8y' is a five-digit number which is divisible by 72, then find the value of (5x - y).
- How many numbers from 1 to 500 (both included) are divisible by both 4 and 6 but not divisible by 8?
Relevant for Exams:
