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practiceMT1

practiceMT1 - 1001117 are divisible by 53 Hint use...

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Practice problems for Midterm I. 1. Find the greatest common divisor of 111111 and 1111. 2. a) Find the last digit of 2 2011 + 3 2011 + 4 2011 . b) Find the remainder of 2 2011 + 3 2011 + 4 2011 modulo 2011. 3. Solve the equations a) 15 x = 6 mod 100, b) 15 x = 5 mod 100. 4. Find the smallest number whose remainder when divided by 12 is 11 and when divided by 15 is 14. 5. Compute a) φ (2 · 3 · 4 · 5 · 6) , b) φ (2011). 6*. Let p be a prime number, and p | a 2 + 1 for some a . Prove that p = 1 mod 4. (Hint: use Fermat’s Theorem). 7. Show that for every positive integer n , 3 2 n - 1 is divisible by 8. 8. Solve the congruence x 3 = 6 mod 7 . 9. Prove that all numbers of the form 10017 , 100117 , 1001117
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Unformatted text preview: , 1001117 ,... are divisible by 53. Hint: use induction, consider the diﬀerence between the consecutive num-bers. 10. Find the compact formula for the sum 1 · 1! + 2 · 2! + 3 · 3! + + n · n ! . Hint: guess a formula for n = 1 , 2 , 3, then prove it by induction. 11. Prove that the remainder of 3 n modulo 20 is less than 10 for all n . 12. Prove that the number (2 k )! /k ! is divisible by 2 k and not divisible by 2 k +1 . Hint: use induction. 13. Prove the following identity for the Fibonacci numbers: F 1 + F 2 + ... + F n = F n +2-1 . Hint: use induction. 1...
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