Department of Mathematics and Statistics

McGill University

189–235A

Class Test

Wednesday, October 25, 1995 8: 35 am. - 9: 25 am.

1. Show that for every integer n with n ≥ 1 the identity

∑_{k=1}^{n} k(k!) = (n + 1)! − 1

holds. The notation m! stands for the factorial ∏_{k=1}^{m} k of m.

2. Suppose that a and b are integers such that a^{2} + b^{2} is divisible by 7. Show that both a and b are divisible by 7.

3. Let R be a ring and let I and J be ideals of R. Let IJ be the set of all finite sums a_{1}b_{1} + a_{2}b_{2} + ⋯ + a_{n}b_{n} with a_{j} ∈ I and b_{j} ∈ J.

(i) Show that IJ is an ideal in R.

(ii) Show that I ∩ J is an ideal in R.

(iii) Show that IJ ⊆ I ∩ J.

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