Unformatted text preview: , 1001117 ,... are divisible by 53. Hint: use induction, consider the diﬀerence between the consecutive numbers. 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 +21 . Hint: use induction. 1...
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 Summer '08
 BESCHER
 Algebra, Division, Equations, Remainder, Natural number, Prime number

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