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# homework5 - THEORY OF NUMBERS Math 115 A Homework 5 Due...

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Unformatted text preview: THEORY OF NUMBERS, Math 115 A Homework 5 Due Friday November 9 1. Find the solutions of each of the following systems of linear congruences: x ≡ (mod 2) x ≡ (mod 3) x ≡ 1 (mod 5) x ≡ 6 (mod 7) 3 x ≡ 5 (mod 2) 5 x ≡ 4 (mod 3) 2 x ≡ 1 (mod 5) 5 x ≡ 6 (mod 7) (Remark: in the second one, rewrite each equation in the form x ≡ a (mod m ) before applying the formula in the Chinese Remainder Theorem) 2. Use the Chinese Remainder Theorem for the moduli 7, 9, 10, 11, and 13 to check whether the following multiplications are correct: 243 · 262 = 64152 , 5023 · 14 = 70322 , 147 · 476 = 83832 . First prove in each case that the product must indeed be smaller than 7 · 9 · 10 · 11 · 13 = 90 , 090. 3. Guess the age of prof. Santos, knowing that it is congruent to 4 modulo 7 and congruent to 7 modulo 4. (Clue: it is not equal to the smallest number with those properties)....
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