c) Alternative proof of patterns using ordp. Let ordp(n) define the largest power d such that p divides n. For every real number z, let Lx] denote the integer part of a and {z) := x-laj denote the fractional part of z. Prove the following properties of ordp i) ordp(n!) = 1 I I 1 + 1 + where I I can be interpreted as the number of positive multiples of p no larger than n ii) ordp(ab) = ords(a) + ordp(b) for any two integers a and b. Use this and (i) to find a formula for ordp) ii) For any two numbers z and y, prove the equivalences: iv) How is { written in terms of the base p form (F.76, of n? Putting this together with the results at (ii) and, try to explain the patterns observed at part (a). c) Alternative proof of patterns using ordp. Let ordp(n) define the largest power d such that p divides n. For every real number z, let Lx] denote the integer part of a and {z) := x-laj denote the fractional part of z. Prove the following properties of ordp i) ordp(n!) = 1 I I 1 + 1 + where I I can be interpreted as the number of positive multiples of p no larger than n ii) ordp(ab) = ords(a) + ordp(b) for any two integers a and b. Use this and (i) to find a formula for ordp) ii) For any two numbers z and y, prove the equivalences: iv) How is { written in terms of the base p form (F.76, of n? Putting this together with the results at (ii) and, try to explain the patterns observed at part (a)