Unit 2 · NumbersN1 · Number Properties · Propulsion

Number Properties

Use factors, multiples, parity, signs, and remainders to reason without brute force.

29 min
Unlocks: Ignition System
Desmos way

Ready when you are

7 steps · about 29 minutes

Start with the warm-up.

Start

Step 1 of 7Warm-upDone

Warm-up

Finish the previous step to continue, or use “Show everything” above to explore.

Key terms

Prime: An integer greater than 1 with exactly two positive factors.
Greatest common factor: The largest factor shared by all given integers.
Least common multiple: The smallest positive multiple shared by all given integers.

Step 2 of 7Try it firstDone

Try it first

Finish the previous step to continue, or use “Show everything” above to explore.

Take a shot before the lesson. Write one relationship from the stem, even if you do not finish.

Step 3 of 7LearnDone

The big idea

Finish the previous step to continue, or use “Show everything” above to explore.

See the structure

Prime factorization is a compact map of a number. The GCF uses the smallest shared exponents; the LCM uses the largest exponents appearing in either number.

Key idea

Name quantities before calculating

Write what each number measures. A correct calculation with the wrong quantity or unit still misses the question.

Build a reliable method

Parity rules save time: even plus even is even, odd plus odd is even, and a product is odd only when every factor is odd.

Explore in Desmos

Check a common factor

Read the GCF 12, then confirm that dividing both 24 and 36 by 12 gives whole numbers. A larger proposed factor must fail for at least one input.

The calculator is live: change anything and see what happens.

Check a common factor

Degree mode

Loading Desmos…

Stretch to a 36

A remainder statement is an equation: dividing $n$ by $d$ with remainder $r$ means $n=dq+r$ , with $0\le r<d$ .

Trap

A believable answer to the wrong question

The choices often include a correct intermediate number. Before selecting it, reread the highlighted ask and check the unit, sign, and domain.

Step 4 of 7Worked examplesDone

Worked examples

Finish the previous step to continue, or use “Show everything” above to explore.

Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1: Build a greatest common factor from primes

Problem

What is the GCF of 84 and 126?

Factor:

84=2^2\cdot3\cdot7

and

126=2\cdot3^2\cdot7

Which factor, multiple, or parity rule applies?

Use shared minimum exponents:

2\cdot3\cdot7=42

Which divisibility or cycle check confirms the result?

Worked example

Example 2: Find when repeating cycles meet

Problem

Signals repeat every 12 and 18 seconds. When do they next coincide?

Prime forms are

12=2^2\cdot3

and

18=2\cdot3^2

Which factor, multiple, or parity rule applies?

Use maximum exponents for an LCM:

2^2\cdot3^2=36

seconds.

Which divisibility or cycle check confirms the result?

Faded example · you finish it

Example 3: Complete the setup and result

Problem

For integer

k

, is

3(2k+1)+5

odd or even?

Choose the setup that follows the stated relationship.

Choose the resulting value.

Desmos way

Sample a parity pattern, then prove it

Type the three mod(...,2) expressions. Each displays 0, meaning the tested value of

3(2k+1)+5

is even.

Sampling supports the pattern but does not prove it for every integer. Algebra finishes the proof:

3(2k+1)+5=6k+8=2(3k+4)

Sample a parity pattern, then prove it

Degree mode

Loading Desmos…

Step 5 of 7PracticeDone

Practice

Finish the previous step to continue, or use “Show everything” above to explore.

Step 6 of 7Spot the mistakeDone

Spot the mistake

Finish the previous step to continue, or use “Show everything” above to explore.

Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

Step 7 of 7Exit ticketDone

Exit ticket

Finish the previous step to continue, or use “Show everything” above to explore.

Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.