Question

Exercise 7 (2 points) Recall the binomial coefficient for integer parameters 0 Sk< n. Prove that Exercise 8 (2 points) Prove the following: if z is an integer with at most three decimal digits aia2a3, then x is divisible by 3 if and only if aut a2 +a3 is divisible by 3. Exercise 9 (3 points) A square number is an integer that is the square of another integer. Let x and y be two integers, each of which can be written as the sum of two square numbers. Show that the product xy can be written as the sum of two square numbers.

Answer 1

Binomial coefficient (n k) is defined as: (n k) = n_C_k = n!/(n - k)! k! consider (n + 1 k + 1) = (n + 1)!/((n + 1) - (k + 1))! (k + 1)! Now consider (n k) + (n k + 1) = n!/(n - k)! k! + n!/(n - (k + 1))! (k + 1)! = n!/(n - k)! k! + n!/(n - k - 1)! (k)! (k + 1) = n!/(n - k - 1)! (n - k) times k! + n!/(n - k - 1)! K! (k + 1) = n!(k + 1) + n!(n - k)/(n - k - 1)! (n - k) times k! (k + 1) = n!(k + 1 + n - k)/(n - k) (n - k - 1)! K! (k + 1) = n! (k - k + n + 1)/(n - k)! (k + 1)! \r\n n! (n + 1)/(n - k + 1 - 1)! (k + 1)!