KMB, 3/12/99

M1F Foundations of Analysis—Problem Sheet 9

This is assessed coursework. Please hand in solutions to the starred questions (that is, 1, 3, 4, 5, 6, at the end of the lecture on Monday 13th December.

1^{∗})

a) Prove that for all integers n ≥ 1, we have 10^{n} ≡ 1 mod 9.

b) Deduce that if a is an integer, then a × 10^{n} ≡ a mod 9.

c) Deduce the “rule of 9”, that is, that a positive integer is a multiple of 9 if and only if the sum of its digits is a multiple of 9.

2)

a) Prove the “rule of 4”, which says that to check that a positive integer is a multiple of 4, it suffices to check to see if the last two digits form a 2-digit number which is a multiple of 4.

b) Prove the “rule of 7”, which says that to check that a positive integer is a multiple of 7, break the number up into three-digit chunks (starting at the “units” end of the number—see example) and alternately add and subtract these numbers, and check instead that this sum is a multiple of 7. For example, to see if the number 26007128 is a multiple of 7, it suffices to work out 26−007+128 = 147 and then to check that 147 is a multiple of 7—which indeed it is, it’s 7 × 21. Hence the rule of 7, once you’ve proved it, will enable you to conclude that 26007128 is also a multiple of 7.

c) Do you think that there is a “rule of n”, for any n?

3^{∗}) Find an integer t between 0 and 6, such that 3^{1999} ≡ t mod 7.

4^{∗}) Proofs or counterexamples required for the following.

a) Is it true that if A, B and C are sets, then

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)?

b) Is it true that if A, B and C are sets, then

(A\B) ∪ (B\C) ∪ (C\A) = A ∪ B ∪ C?

5^{∗}) In a typical M1F lecture there seem to about 150 students. Let’s say that during a typical Monday morning lecture, 20 of the students present are eating their breakfast, and 30 are reading the paper. Everyone who is neither eating nor reading is asleep in their chair. If 115 students are asleep, how many are both eating breakfast and reading the paper simultaneously?

6^{∗}) For n ≥ 1 an integer, let A_{n} denote the set \{x ∈ ℝ : \frac{1}{n} < x < \sqrt{2} + \frac{1}{n}\}. Work out (with proof) ⋃_{n=1}^{∞} A_{n} and ⋂_{n=1}^{∞} A_{n}.