The Fundamental Theorem of Arithmetic

Try this

Each press of $+$ multiplies the number by that prime. Try setting $a$ to $100$ , then to $1$ . Can two different sets of exponents give the same number?
Click “gcd: take the lower”. Under each prime, which blocks of which bar are filled in? Multiplied together, they give $(a, b)$ .
Click “lcm: take the higher”. What is filled in now?
Look at the two products on the right: $(a, b) \cdot [a, b]$ and $a \cdot b$ are always equal. Why, judging from the bars? (In each group, lower plus higher equals the two bars added together.)
Make every bar of $b$ no taller than the matching bar of $a$ . What is $(a, b)$ now? How are $b$ and $a$ related?

Fundamental Theorem of Arithmetic Every integer greater than $1$ can be written as a product of primes, and apart from the order of the prime factors, there is only one way to do it.

Collect equal primes together and you get the prime factorization $n = p_{1}^{a_{1}} p_{2}^{a_{2}} \dots p_{k}^{a_{k}}$ . That string of exponents is the number's one-of-a-kind barcode. So:

d ∣ n ⟺ each exponent of d is at most the matching exponent of n

Taking the $min$ for the gcd and the $max$ for the lcm both follow directly from this one statement.

Predict first

$72 = 2^{3} \times 3^{2}$ . How many positive divisors does it have?

Choose an answer first; then you can see the result.

Why

How many divisors does 72 have?

Step 1 / 5

What the divisors look like
$72 = 2^{3} \times 3^{2}$ . Its divisors can only contain the prime factors $2$ and $3$ , and the exponents cannot go over: $d = 2^{i} \times 3^{j}$ with $0 \leq i \leq 3$ and $0 \leq j \leq 2$ .

$d = 2^{i} \times 3^{j}$
Walk along ×2
Walk along ×3
The two decisions do not interfere
Dividing means “lying below”

Remove one condition

“There is only one way to write it” sounds obvious. In fact it depends on multiplying within all the positive integers.

Break it

Counterexample lab: remove one condition

Conclusion: breaking a number into numbers that “cannot be split further” gives only one result

In the ordinary positive integers, however you break up $60$ , at the bottom you always get $2 \times 2 \times 3 \times 5$ . In the divisor lattice, every path from $1$ up to $60$ uses the same number of lines of each of the three kinds.

Practice

Exercise 1Basic

$1400 = 2^{a} \times 5^{b} \times 7^{c}$ . Find $a + b + c$ .

Exercise 2Basic

How many positive divisors does $360$ have?

Exercise 3Advanced

What is the smallest positive integer with exactly $10$ positive divisors?

Exercise 4Challenge

Find the smallest positive integer $n$ such that $360 n$ is a perfect cube.

Try this

Why

How many divisors does 72 have?

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Adding the “exponent plus one” numbers to count divisors

Taking the lcm to be the product