Famous Theorems of Mathematics/Number Theory/Basic Results (Divisibility)

Definition of divisibility

An integer b is divisible by an integer a, not zero, if there exists an integer x such that b = ax and we write $a|b$. In case b is not divisible by a, we write $a \nmid b$. For example $3|6$ and $4 \nmid 6$.

If $a | b$ and 0 < a < b then a is called a proper divisor of b. The notation $a^k || b$ is used to indicate that $a^k | b$ but $a^{k+1} \nmid b$. For example $3||6$.

Basic properties

1. $a|b$ implies $a|bc$ for any integer c.
2. $a|b$ and $b|c$ imply $a|c$.
3. $a|b$ and $a|c$ imply $a|(bx + cy)$ for any integers x and y.
4. $a|b$ and $b|a$ imply $a=\pm b$.
5. $a|b$, a > 0, b > 0, imply $a \le b$.
6. If $m \ne 0$, $a|b$ implies and is implied by $ma|mb$.

Proof:

1. Clearly if $a | b$ then $\exists$ x such that b = ax. Then bc = a(xc) = ay and so $a | bc$.
2. $a | b$ and $b | c$ imply that m,n exist such that b = am and c = bn. Then clearly c = bn = (am)n = ap and so $a | c$.
3. $a | b$ and $a | c$ imply existence of m,n such that b = am and c = an so that bx + cy = amx + any = az and so $a | bx + cy$.
4. $a | b$ and $b | a$ imply that m,n exist such that b = am and a = bn. Then a = bn = amn and so mn = 1. The only choice for m and n is to be simultaneously 1 or -1. In either case we have the result.
5. b = ax for some x > 0 as otherwise a and b would have opposite signs. So b = (a + a + ...) x times $\ge$ a.
6. $a | b$ implies b = ax for some x which imples mb = amx i.e. $ma | mb$. Conversely $ma | mb$ implies mb = max from which b = ax and so $a | b$ follows.

Remarks:

• Properties 2 and 3 can be extended by the principle of mathematical induction to any finite set. That is, $a_1 | a_2$, $a_2 | a_3$, $a_3 | a_4 \cdots a_{k-1} | a_k$ implies $a_1 | a_k$; and $a | b_1$, $a | b_2 \cdots a | b_n$ implies that $a | \sum_{j=1}^n b_jx _j$ for any integers $x_j$.
• Property 4 uses the fact that the only units in the set of integers are 1 and -1. (A unit in a ring is an element which possesses a multiplicative inverse.)

The division algorithm

Specifically, the division algorithm states that given two integers a and d, with d ≠ 0

There exist unique integers q and r such that a = qd + r and 0 ≤ r < | d |, where | d | denotes the absolute value of d.

Note: The integer

• q is called the quotient
• r is called the remainder
• d is called the divisor
• a is called the dividend

Proof: The proof consists of two parts — first, the proof of the existence of q and r, and secondly, the proof of the uniqueness of q and r.

Let us first consider existence.

Consider the set

$S = \left\{a - nd : n \in \mathbb{Z}\right\}$

We claim that S contains at least one nonnegative integer. There are two cases to consider.

• If d < 0, then −d > 0, and by the Archimedean property, there is a nonnegative integer n such that (−d)n ≥ −a, i.e. a − dn ≥ 0.
• If d > 0, then again by the Archimedean property, there is a nonnegative integer n such that dn ≥ −a, i.e. a − d(−n) = a + dn ≥ 0.

In either case, we have shown that S contains a nonnegative integer. This means we can apply the well-ordering principle, and deduce that S contains a least nonnegative integer r. If we now let q = (a − r)/d, then q and r are integers and a = qd + r.

It only remains to show that 0 ≤ r < |d|. The first inequality holds because of the choice of r as a nonnegative integer. To show the last (strict) inequality, suppose that r ≥ |d|. Since d ≠ 0, r > 0, and again d > 0 or d < 0.

• If d > 0, then r ≥ d implies a-qd ≥ d. This implies that a-qd-d ≥0, further implying that a-(q+1)d ≥ 0. Therefore, a-(q+1)d is in S and, since a-(q+1)d=r-d with d>0 we know a-(q+1)d<r, contradicting the assumption that r was the least nonnegative element of S.

• If d<0 then r ≥ -d implying that a-qd ≥ -d. This implies that a-qd+d ≥0, further implying that a-(q-1)d ≥ 0. Therefore, a-(q-1)d is in S and, since a-(q-1)d=r+d with d<0 we know a-(q-1)d<r, contradicting the assumption that r was the least nonnegative element of S.

In either case, we have shown that r > 0 was not really the least nonnegative integer in S, after all. This is a contradiction, and so we must have r < |d|. This completes the proof of the existence of q and r.

Now let us consider uniqueness.

Suppose there exists q, q' , r, r' with 0 ≤ r, r' < |d| such that a = dq + r and a = dq' + r' . Without loss of generality we may assume that q ≤ q' .

Subtracting the two equations yields: d(q' - q) = (r - r' ).

If d > 0 then r' ≤ r and r < d ≤ d+r' , and so (r-r' ) < d. Similarly, if d < 0 then r ≤ r' and r' < -d ≤ -d+r, and so -(r- r' ) < -d. Combining these yields |r- r' | < |d|.

The original equation implies that |d| divides |r- r' |; therefore either |d| ≤ |r- 'r' | (if |r- r' | > 0 so that |d| is also > 0 and property 5 of Basic Properties above holds), or |r- r' |=0. Because we just established that |r-r' | < |d|, we may conclude that the first possibility cannot hold. Thus, r=r' .

Substituting this into the original two equations quickly yields dq = dq' and, since we assumed d is not 0, it must be the case that q = q' proving uniqueness.

Remarks:

• The name division algorithm is something of a misnomer, as it is a theorem, not an algorithm, i.e. a well-defined procedure for achieving a specific task.
• There is nothing particularly special about the set of remainders {0, 1, ..., |d| − 1}. We could use any set of |d| integers, such that every integer is congruent to one of the integers in the set. This particular set of remainders is very convenient, but it is not the only choice.