Prime-factor products

Which number is prime?

A factor is not always prime

A factor divides a whole number exactly. A prime is greater than one and has exactly two positive whole-number factors: one and itself. A composite is greater than one and has more than two

A composite can split

Twelve has factor pairs one and twelve, two and six, three and four

A prime cannot split into smaller whole-number factors

The primes up to thirteen are two, three, five, seven, eleven and thirteen. Oddness alone is not enough: nine has a factor three

One is neither

One has only one positive factor. Zero and one are not prime or composite, and one is not a prime factor to add to a finished product

An exponent counts equal factors

The base is the repeated factor. A positive whole-number exponent tells you how many copies to multiply. It does not tell you to multiply the base by the exponent

Base two

Four copies of two make sixteen

Base three

Three copies of three make twenty-seven

Base five

Two copies of five make twenty-five

The power sequence reaches one

Reduce the exponent by one and divide the value by the base. After exponent one, dividing once more gives one, not zero

Divide by two each step

Four becomes two, then one

Divide by three each step

Nine becomes three, then one

Divide by five each step

Twenty-five becomes five, then one. This does not make one a prime factor

Evaluate a zero power

Split every composite branch

A factor tree records a product at each split. Stop a branch only when its number is prime. Different first splits can reach the same prime factors

One first split

Three is prime. Six still splits into two and three

Another first split

Two is prime. Nine still splits into three and three

Read the tree

Read from left to right. An uncircled composite splits; a circled prime leaf does not. Multiply all the circled leaves once. Do not also multiply their parent numbers

This tree starts at eighteen. Predict what the tree will show if the starting number changes to nineteen

Change eighteen to nineteen and observe the prime tree. Then try sixteen and thirty-six. Follow every split and multiply only the circled leaves. The tool shows one valid tree, not every possible first split

Set the number to thirty-six. What is the product of its circled leaves?

One power for each repeated prime

Count how often each prime appears among the leaves. Keep one base for that prime and use its count as the exponent

Group the repeats

There are two twos and two threes. The factors may be reordered before grouping

A prime used once

A single factor can be bare or have exponent one. Do not add a factor one or keep a composite base

Enter the grouped form

Use * or × between factors and ^ for a power. Any factor order is allowed. A scalar answer has the same value but does not show the requested factorisation

Group supplied prime factors

Another method: divide by primes

To factor a number without a supplied tree, try small prime divisors. Record each divisor, including repeats. Stop when the remaining quotient is one

Divide until the quotient is one

Eighty-four is even, so start with two. Keep each successful prime divisor

Collect and group the divisors

The divisors are two, two, three and seven. Group only the repeated two

Check the product

Two squared is four. Four times three times seven returns eighty-four. The final quotient one is not an extra prime divisor

Construct a complete prime product

Evaluate powers before multiplying

A product in index form can be read back to its number. Work out each power, then multiply those values

Work out the powers

Two cubed is eight and five squared is twenty-five

Multiply their values

Evaluate a prime product

Same value does not always mean finished

Check three things: does the product have the target value, is every base prime, and is each repeated prime grouped into one power?

Complete prime-factor form

This has value forty-eight, prime bases and grouped repeats. Reversing the factors is also valid

Right value, composite base

This also has value forty-eight, but twelve is composite. The factorisation is not finished

Right value, ungrouped repeat

These factors also multiply to forty-eight, but two occurs in two separate factors. Combine them into one power of two

Wrong value

Removing one factor of two gives twenty-four, not forty-eight

Check both proposed products

Keep the value and finish the factorisation

A complete prime-factor product preserves the number while showing its prime building blocks

Construct

Split composites or divide by primes, record every prime factor, then group repeats into powers

Evaluate

Work out each power, then multiply. For the positive prime bases used here, exponent zero gives one

Check

Use the target value, prime bases and grouped repeats. Order does not matter, but an unfinished product is not the requested form

A complete worked factorisation

Write the complete prime factorisation in index form

Divide by primes and record every divisor, stopping at quotient one

The recorded prime divisors multiply back to the starting number

Three twos and two threes become the exponents. One is not an extra prime factor