Integers, divisors, and primes

8.1 Check (using the definition) that $1\mid a$, $-1\mid a$, $a\mid a$ and $-a\mid a$ for every integer $a$.

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8.2 What does it mean for $a$, in more everyday terms, if (a) $2\mid a$; (b) $2\nmid a$; (c) $0\mid a$.

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8.3 Prove that

(a) if $a\mid b$ and $b\mid c$ then $a\mid c$;

(b) if $a\mid b$ and $a\mid c$ then $a\mid b+c$ and $a\mid b-c$;

(c) if $a,b>0$ and $a\mid b$ then $a\le b$;

(d) if $a\mid b$ and $b\mid a$ then either $a=b$ or $a=-b$.

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8.4 Let $r$ be the remainder of the division $b:a$. Assume that $c\mid a$ and $c\mid b$. Prove that $c\mid r$.

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8.5 Assume that $a\mid b$, and $a,b>0$. Let $r$ be the remainder of the division $c:a$, and let $s$ be the remainder of the division $c:b$. What is the remainder of the division $s:a$?

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8.6 (a) Prove that for every integer $a$, $a-1\mid a^2-1$.

(b) More generally, for every integer $a$ and positive integer $n$,

$$a-1\mid a^n-1.$$

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8.7 Are there any even primes?

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8.8 (a) Prove that if $p$ is a prime, $a$ and $b$ are integers, and $p\mid ab$, then either $p\mid a$ or $p\mid b$ (or both).

(b) Suppose that $a$ and $b$ are integers and $a\mid b$. Also suppose that $p$ is a prime and $p\mid b$ but $p\nmid a$. Prove that $p$ is a divisor of the ratio $b/a$.

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8.9 Prove that the prime factorization of a number $n$ contains at most ${\log }_{2}n$ factors.

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8.10 Let $p$ be a prime and $1\le a\le p-1$. Consider the numbers $a$, $2a$, $3a$, $\dots$, $(p-1)a$. Divide each of them by $p$, to get residues $r_1,r_2,\dots ,r_{p-1}$. Prove that every integer from $1$ to $p-1$ occurs exactly once among these residues.

[Hint: First prove that no residue can occur twice.]

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8.11 Prove that if $p$ is a prime, then $\sqrt{p}$ is irrational. More generally, prove that if $n$ is an integer that is not a square, then $\sqrt{n}$ is irrational.

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8.12 Try to formulate and prove an even more general theorem about the irrationality of the numbers $\sqrt[k]{n}$.

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8.13 Show that among $k$-digit numbers, one in about every $2.3k$ is a prime.

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8.14 Show by examples that neither the assertion in lemma 8.1 nor Fermat’s Little Theorem remain valid if we drop the assumption that $p$ is a prime.

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8.15 Consider a regular $p$-gon, and all $k$-subsets of the set of its vertices, where $1\le k\le p-1$. Put all these $k$-subsets into a number of boxes: we put two $k$-subsets into the same box if they can be rotated into each other. For example, all $k$-subsets consisting of $k$ consecutive vertices will belong to one and the same box.

(a) Prove that if $p$ is a prime, then each box will contain exactly $p$ of these sets.

(b) Show by an example that (a) does not remain true if we drop the assumption that $p$ is a prime.

(c) Use (a) to give a new proof of Lemma 8.1.

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8.16 Imagine numbers written in base $a$, with at most $p$ digits. Put two numbers in the same box if they arise by cyclic shift from each other. How many will be in each class? Give a new proof of Fermat’s theorem this way.

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8.17 Give a third proof of Fermat’s “Little Theorem” based on exercise 8.3.

[Hint: consider the product $a(2a)(3a)\dots ((p-1)a)$.]

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8.18 Show that if $a$ and $b$ are positive integers with $a\mid b$, then $\gcd (a,b)=a$.

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8.19 (a) $\gcd (a,b)=\gcd (a,b-a)$.

(b) Let $r$ be the remainder if we divide $b$ by $a$. Then $\gcd (a,b)=\gcd (a,r)$.

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8.20 (a) If $a$ is even and $b$ is odd, then $\gcd (a,b)=\gcd (a/2,b)$.

(b) If both $a$ and $b$ are even, then $\gcd (a,b)=2\gcd (a/2,b/2)$.

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8.21 How can you express the least common multiple of two integers, if you know the prime factorization of each?

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8.22 Suppose that you are given two integers, and you know the prime factorization of one of them. Describe a way of computing the greatest common divisor of these numbers.

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8.23 Prove that for any two integers $a$ and $b$,

$$\gcd (a,b)\mathrm{lcm}(a,b)=ab.$$

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8.24 Show that the euclidean algorithm can terminate in two steps for arbitrarily large positive integers, even if their g.c.d. is 1.

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8.25 Describe the euclidean algorithm applied to two consecutive Fibonacci numbers. Use your description to show that the euclidean algorithm can last arbitrarily many steps.

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8.26 Suppose that $a<b$ and the euclidean algorithm applied to $a$ and $b$ takes $k$ steps. Prove that $a\ge F_k$ and $b\ge F_{k+1}$.

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8.27 Consider the following version of the euclidean algorithm to compute $\gcd (a,b)$:

(1) swap the numbers if necessary to have $a\le b$; (2) if $a=0$, then return $b$; (3) if $a\ne 0$, then replace $b$ by $b-a$ and go to (1).

(a) Carry out this algorithm to compute $\gcd (19,2)$.

(b) Show that the modified euclidean algorithm always terminates with the right answer.

(c) How long does this algorithm take, in the worst case, when applied to two 100-digit integers?

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8.28 Consider the following version of the euclidean algorithm to compute $\gcd (a,b)$.

Start with computing the largest power of 2 dividing both $a$ and $b$. If this is $2^r$, then divide $a$ and $b$ by $2^r$. After this “preprocessing”, do the following:

(1) Swap the numbers if necessary to have $a\le b$.

(2) If $a\ne 0$, then check the parities of $a$ and $b$; if $a$ is even, and $b$ is odd, then replace $a$ by $a/2$; if both $a$ and $b$ are odd, then replace $b$ by $b-a$; in each case, go to (1).

(3) If $a=0$, then return $2^{r}b$ as the g.c.d.

Now come the exercises:

(a) Carry out this algorithm to compute $\gcd (19,2)$.

(b) It seems that in step (2), we ignored the case when both $a$ and $b$ are even. Show that this never occurs.

(c) Show that the modified euclidean algorithm always terminates with the right answer.

(d) Show that this algorithm, when applied to two 100-digit integers, does not take more than 1500 iterations.

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8.29 Show that if $n$ has $k$ bits in base 2, then $2^n$ can be computed using less than $2k$ multiplications.

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8.30 Show that $561\mid a^{561}-a$, for every integer $a$.

[Hint: since $561=3\cdot 11\cdot 17$, it suffices to prove that $3\mid a^{561}-a$, $11\mid a^{561}-a$ and $17\mid a^{561}-a$. Prove these relations separately, using the method of the proof of the fact that $341\mid 2^{340}-1$.]

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