Question

78 CHAPTER 5. THE CHINESE REMAINDER THEOREM 6. (a) Let m mi,m2 Then r a (mod mi), ag (mod m2) can be solved if and only if (m, m2) | a1-a2. The solution, when it exists, is unique modulo m. (b) Using part (a) prove the Chinese remainder theorem by induction. 7. There is a number. It has no remainder when divided by five. It has a remainder ten when divided by seven hundred and fifteen. It has a remainder one hundred and forty when divided by two hundred and forty seven. It has a remainder two hundred and forty five when divided by three hundred and ninety has a remainder one hundred and nine when divided by one hundred and eighty seven. May we ask what is the number? one. It 8. Given positive integers d and m with d m and a any number with (a, d) = 1. Prove that one can find a number a' with (a', m) 1 and a' a (mod d). 9. Find the last three digits in the largest known prime. 10. (Problem #8 APICS 2001 Contest) Find 3 consecutive integers, the first being a multiple of the square of a prime number, the second being a multiple of the cube of a prime number and the last being a multiple of the fourth power of a prime number.

Please help me with understandable solutions for question 6(a), 7, 8 and 10. ( Use Chinese remainder theorem where applicable).

Answer 1

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