Prime Factors, HCF & LCM Flashcards

All 23 cards in this deck

What is a prime number?

A number with exactly two factors: 1 and itself.
e.g. $2, 3, 5, 7, 11$

True or false? 1 is a prime number.

False. 1 has only one factor, so it is not prime; the smallest prime is 2.

What is a factor (divisor) of a number?

A whole number that divides into it exactly, with no remainder.
e.g. factors of 12: $1, 2, 3, 4, 6, 12$

What is a multiple of a number?

The result of multiplying that number by a whole number.
e.g. multiples of 6: $6, 12, 18, 24, \dots$

True or false? 2 is the only even prime number.

True. Every other even number has 2 as a factor as well as 1 and itself.

What are the steps to list all the factors of a number?
e.g. list the factors of 24

1. Start with the pair $1 \times 24$.
2. Test $2, 3, 4, \dots$ in turn: $2 \times 12$, $3 \times 8$, $4 \times 6$.
3. Stop when the pairs meet and write them in order: $1, 2, 3, 4, 6, 8, 12, 24$.

What does it mean to write a number as a product of its prime factors?

Writing it as a multiplication in which every factor is prime.
e.g. $60 = 2 \times 2 \times 3 \times 5$

What are the steps of the factor tree method for prime factorisation?
e.g. 36

1. Split the number into any factor pair: $36 = 4 \times 9$.
2. Keep splitting any non-prime branch: $4 = 2 \times 2$, $9 = 3 \times 3$.
3. Multiply all the circled primes: $36 = 2^2 \times 3^2$.

What are the steps of the repeated division method for prime factorisation?
e.g. 90

1. Divide by the smallest prime that goes in: $90 \div 2 = 45$.
2. Repeat on each answer: $45 \div 3 = 15$, $15 \div 3 = 5$, $5 \div 5 = 1$.
3. Multiply the divisors used: $90 = 2 \times 3^2 \times 5$.

How do you write a prime factorisation in index (product) notation?
e.g. $2 \times 2 \times 2 \times 7$

Collect repeated primes as powers, writing primes in increasing order: $2^3 \times 7$.

What does the unique factorisation theorem state?

Every integer greater than 1 can be written as a product of primes in exactly one way (apart from the order of the factors).

True or false? Starting a factor tree for 36 with $4 \times 9$ instead of $6 \times 6$ gives a different prime factorisation.

False. Any correct starting pair leads to the same set of primes, $2^2 \times 3^2$.

How can you tell from its prime factorisation that a number is a square number?

Every prime appears to an even power.
e.g. $900 = 2^2 \times 3^2 \times 5^2$

How can you tell from its prime factorisation that a number is a cube number?

Every index is a multiple of 3.
e.g. $216 = 2^3 \times 3^3$

What are the steps to find the smallest integer to multiply a number by to make a square number?
e.g. 40

1. Write it as a product of prime powers: $40 = 2^3 \times 5$.
2. Pick out the primes with odd indices: $2$ and $5$.
3. Multiply those primes together: multiply 40 by $2 \times 5 = 10$.

How do you find the square root of a number from its prime factorisation?
e.g. $2^4 \times 3^2$

Halve each index.
$\sqrt{2^4 \times 3^2} = 2^2 \times 3 = 12$

True or false? A number whose prime factorisation contains a prime raised to an odd power cannot be a square number.

True. A square number has every prime to an even power.

What is the highest common factor (HCF) of two numbers?

The largest number that is a factor of both of them.
e.g. HCF of 60 and 84 is 12

What is the lowest common multiple (LCM) of two numbers?

The smallest number that is a multiple of both of them.
e.g. LCM of 24 and 40 is 120

How do you find the HCF from prime factorisations in index form?
e.g. $A = 2^3 \times 3 \times 5$, $B = 2^2 \times 3 \times 5^2$

Multiply the lowest power of each prime that appears in both.
HCF $= 2^2 \times 3 \times 5 = 60$

How do you find the LCM from prime factorisations in index form?
e.g. $A = 2^3 \times 3 \times 5$, $B = 2^2 \times 3 \times 5^2$

Multiply the highest power of every prime that appears in either.
LCM $= 2^3 \times 3 \times 5^2 = 600$

In a word problem, when do you need the LCM and when do you need the HCF?

LCM for repeating events happening together or a smallest possible total; HCF for splitting into the largest equal groups or largest identical tiles.

True or false? The LCM of two numbers is always the two numbers multiplied together.

False. HCF $\times$ LCM $=$ product of the two numbers, so this only happens when the HCF is 1.