Unit 5: Whole Numbers

Factors come in pairs

A factor of 12 is a number that divides it exactly. They arrive two at a time: 1 × 12, 2 × 6, 3 × 4.

So 12 has six factors: 1, 2, 3, 4, 6, 12.

Hunting in pairs is what stops you missing one. You only have to check up to where the pair meets in the middle.

Quick tests for dividing

You can often tell without dividing at all.

By 2: it ends in an even digit.
By 5: it ends in 0 or 5.
By 3: add the digits up — if that total divides by 3, so does the number.
By 9: the same trick, with 9.

So 342 divides by 3, because 3 + 4 + 2 = 9.

Numbers that will not break

A prime has exactly two factors: 1 and itself. 2, 3, 5, 7, 11, 13, and on forever.

1 is not prime — it has only *one* factor, so it misses the rule by a whisker. That is not fussiness: if 1 counted, the next idea would fall apart.

And 2 is the only even prime, because every other even number has 2 as a spare factor.

Every number is made of primes

Take any whole number and keep splitting it until you cannot.

60 = 6 × 10 = 2 × 3 × 2 × 5, which tidies up to 2^2 × 3 × 5.

Here is the remarkable part: it does not matter where you start. Split 60 any way you like and you always land on exactly those primes. Every number has one and only one prime recipe.

Primes tell you the factors

Once you know 12 = 2^2 × 3, you know how many factors 12 has without listing them.

You may take none, one or two 2s (3 choices) and none or one 3 (2 choices). That is 3 × 2 = 6 factors, and 12 does indeed have six.

Add one to each exponent, then multiply. That is the whole method.

Build new tests from old ones

You already have tests for 2, 3, 5 and 9. Two more, and then a way to make the rest yourself.

By 4: look at the last two digits. If they make a number that divides by 4, so does the whole thing — because 100 divides by 4, so everything above the tens is safe.

By 8: the last three digits, for the same reason.

Combining tests

A number divides by 6 if it divides by both 2 and 3 — because 6 is 2 × 3 and they share no factors.

The same trick builds a test for 12 (4 and 3), or 15 (3 and 5).

But it fails for 8: dividing by 2 and by 4 is not enough, because 2 and 4 share a factor. 12 passes both and does not divide by 8.

The strange one: eleven

For 11, add the digits alternately plus and minus.

For 918082: 9 - 1 + 8 - 0 + 8 - 2 = 22. That divides by 11, so 918082 does too.

It looks like magic and it is not: powers of ten alternate between one more and one less than a multiple of 11.

Factors come in pairs — except sometimes

Factors arrive two at a time. So every number should have an even count of them.

But 16 has five: 1, 2, 4, 8, 16. What went wrong?

Nothing. 4 × 4 is a pair where both halves are the same number, so it only gets written once. A number has an odd number of factors exactly when it is a perfect square.

The lockers puzzle

A hundred lockers, all shut. Student 1 flips every locker. Student 2 flips every 2nd. Student 3 every 3rd, and so on to 100.

A locker gets flipped once for each of its factors. So it ends up open only if it has an odd number of factors.

Which means only the perfect squares are open: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Ten lockers, from a fact about pairs.

Why two minuses make a plus

Look at the pattern and let it finish itself:

3 × -2 = -6, 2 × -2 = -4, 1 × -2 = -2, 0 × -2 = 0.

Each step adds 2. Keep going: -1 × -2 must be +2.

Nobody decided that. It is the only answer that keeps the pattern from breaking.

Counting the minuses

When several negatives are multiplied, only how many there are matters.

An even number of minus signs gives a positive. An odd number gives a negative.

So -2 × -3 × -4 is negative — three minuses — and comes to -24.

Only the end matters

What is the last digit of 7^100? Nobody is going to work out that number.

But the last digit of a product depends only on the last digits being multiplied. So the ends go in a cycle:

7, 9, 3, 1, then 7 again. Four steps, exactly like the powers of i.

Some cycles are very short

Powers of 5 always end in 5. Powers of 6 always end in 6. Those cycles have length one.

Powers of 4 alternate 4, 6, 4, 6 — a cycle of two.

So the first thing to do is find the cycle. Everything after that is a remainder.

The biggest thing they share

The greatest common divisor of two numbers is the largest number that goes into both.

For 24 and 36, the shared factors are 1, 2, 3, 4, 6, 12 — so the gcd is 12.

The prime recipes give it instantly: 24 = 2^3 × 3 and 36 = 2^2 × 3^2. Take the *smaller* count of each shared prime: two 2s and one 3, which is 12.

The smallest thing they both reach

The lowest common multiple is the first number they both count to. For 4 and 6 it is 12.

That is the one you need when adding 1/4 + 1/6: both fractions have to be rewritten over 12 before they can be added at all.

And a neat fact worth keeping: gcd × lcm always equals the two numbers multiplied together.

Remainders go round in circles

Divide 17 by 5 and you get 3 with 2 left over. We say 17 leaves a remainder of 2.

Remainders repeat. Dividing by 5, every number leaves 0, 1, 2, 3 or 4 — round and round forever. That is why you can answer "what day is it in 100 days?" without counting: only the remainder after dividing by 7 matters.