Divisibility and primes: the internal structure of a number

Where we are on the map

We investigate integers through their multiplicative structure.

Begin with a familiar image

Every way to arrange 84 objects into equal groups identifies a factor. A factor tree opens a composite number until only primes remain at its leaves.

Precise meaning

If a divides by b with no remainder, b is a factor and a is a multiple of b. A prime number has exactly two positive factors: 1 and itself.

The lesson’s support signal

meaning → model → calculation → check → explain in your own words

factor pairs → prime factors → multiply back to check

Worked example

84 = 4 × 21 = (2 × 2) × (3 × 7) = 2² × 3 × 7.

Example with a fading prompt

Change the numbers in the example and repeat the solution without copying its steps. Estimate first, calculate exactly, and check against the original condition.

Recall without a prompt

1. Define the main idea in one sentence.
2. Rebuild the support signal from memory.
3. Solve the example with different numbers and explain why every step is valid.

Find and correct the mistake

Calling 1 a prime number is wrong: a prime needs two distinct positive factors, while 1 has only one.

Transfer to a new setting

Apply the same relationship to something you can measure yourself. Name the quantities, units, and the boundary within which the model is valid. Check the answer by a second representation or an inverse operation.

Exercise

Required. Factor 90 and 126 into primes, multiply back to check, and find their greatest common factor.

With your own data. Create one everyday example with the same structure and show it in words, a formula, and a check.

Optional. Seven days later, change the numbers and repeat without the support map.

Open perspective

Decimals extend the place-value system to parts smaller than one.

Course contents