Factors, multiples and primes

Write down an odd number between 10 and 20.

Write down two cube numbers less than 10.

Here are all the factors of a number, A.

1 2 3 5 6 10 15 30

Write down the value of A.

Here is a number, B, written as the product of its prime factors.

B = 3 × 5 × 7

Find the highest common factor (HCF) of A and B.

Find a number between 150 and 200 which has factors 7 and 8.

Here is a list of numbers.

15 17 21 28 39 49

From this list, write down

the prime number,

the square number,

two numbers with a difference of 18.

Sara and Tom both make a sequence of numbers. The first term in both sequences is 10.

The term-to-term rule for Sara’s sequence is add 14. The term-to-term rule for Tom’s sequence is add 35.

Each sequence stops when a term reaches or goes above 500.

Find the number of terms that are the same in both Sara’s and Tom’s sequence. You must show your working.

Find an expression for the n th term of Sara’s sequence.

Sara and Tom both make a sequence of numbers. The first term in both sequences is 10.

The term-to-term rule for Sara’s sequence is add 14. The term-to-term rule for Tom’s sequence is add 35.

Each sequence stops when a term reaches or goes above 500.

Find the number of terms that are the same in both Sara’s and Tom’s sequence. You must show your working.

Find an expression for the n th term of Sara’s sequence.

A fair four-sided dice is numbered 3, 4, 5 and 6. The dice is rolled twice and the two scores are multiplied together.

Complete the table to show all the possible outcomes.

Find the probability that the outcome is a multiple of 8.

16 807 = 7⁵.

Write 16 807 000 000 as a product of its prime factors.

Write down an odd number.

Write down a common multiple of 3 and 17.

Write 24: 60 as a ratio in its simplest form.

N is a number such that:

• N = 3 × 5 × k, where k is a prime number

• N is greater than 400.

Find the smallest possible value of N.

a and b are different prime numbers.

Explain why a × b is not a prime number.

Write down a multiple of 9 between 30 and 40.

Write down a factor of 100 between 11 and 30.

Rosa is thinking of a fraction. The numerator is a cube number less than 100. The denominator is a square number less than 100.

Find the fraction that Rosa is thinking of.

Write down all the factors of 15.

Find the largest number that will divide exactly into 15 and 60.

Sasha has these two sets of number cards.

One card is taken at random from each set. Sasha adds the numbers on the two cards to get a total.

Complete the table to show all the possible totals.

Find the probability that the total is a prime number. Give your answer as a fraction.

N is a number such that:

• N = 3 × 5 × k, where k is a prime number

• N is greater than 400.

Find the smallest possible value of N.

a and b are different prime numbers.

Explain why a × b is not a prime number.

Write down an example of each of the following.

An even number.

A multiple of 7.

A cube number between 20 and 220.

A prime number less than 10.

Kai has these four number cards.

Kai takes two of the cards at random without replacement and finds the positive difference between the two numbers.

Complete the table to show all of the possible differences.

Find the probability that Kai takes two cards with a difference that is an even number or a factor of 10.

Two numbers, A and B, are written as the product of their prime factors.

Find the lowest common multiple (LCM) of A and B. Give your answer as an ordinary number.

A number, R, is written as the product of its prime factors.

R = 2 × 3² × 5 × k, where k is a prime number.

The highest common factor (HCF) of R and another number, P, is 26.

Find the value of k.

Kai has these four number cards.

Kai takes two of the cards at random without replacement and finds the positive difference between the two numbers.

Complete the table to show all of the possible differences.

Find the probability that Kai takes two cards with a difference that is an even number or a factor of 10.

Two numbers, A and B, are written as the product of their prime factors.

Find the lowest common multiple (LCM) of A and B. Give your answer as an ordinary number.

A number, R, is written as the product of its prime factors.

R = 2 × 3² × 5 × k, where k is a prime number.

The highest common factor (HCF) of R and another number, P, is 26.

Find the value of k.

Write down the three factors of 25.

Write down a square number between 40 and 70.

A prime number is a whole number that has exactly two factors.

Explain why 1 is not a prime number.

a and b are prime numbers. Write down the 6 factors of a²b.

Riley and Sam are conducting surveys. They are both given the same list of 12 463 people from which to select their sample.

Riley selects every 56th person. Sam selects every 64th person. They both start counting from the first name in the list.

Work out how many people will be selected to be in both surveys. You must show your working.

In the number 34 752, the digit 4 represents four thousand.

Write in words what the digit 7 represents.

Write eight million in figures.

What type of numbers are 2, 3 and 5?

Circle one answer from the list.

cube numbers even numbers odd numbers prime numbers square numbers

These are five tiles.

Arrange the five tiles to make a calculation with the answer 3.

Write down a multiple of 8 that can be made using two of the five tiles.

A prime number is a whole number that has exactly two factors.

Explain why 1 is not a prime number.

a and b are prime numbers. Write down the 6 factors of a²b.

Riley and Sam are conducting surveys. They are both given the same list of 12 463 people from which to select their sample.

Riley selects every 56th person. Sam selects every 64th person. They both start counting from the first name in the list.

Work out how many people will be selected to be in both surveys. You must show your working.

Write 60 as a product of its prime factors.

Here is a list of numbers.

8 11 19 26 39 49 65 114

From this list, write down

an even number,

a square number,

a factor of 57.

Kai has four differently numbered cards.

• The range of the numbers is 14.

• The median of the numbers is 9.

• All the numbers are prime numbers.

• The lowest number is 5.

Work out the numbers on the cards. Write the numbers in order of size.

........,........,........,........ 5

lowest

Trams to the airport leave every 50 minutes. Trams to the beach leave every 35 minutes. A tram to the airport and a tram to the beach leave together at 9:30 am.

When is the next time that two of these trams leave together?

Show that 95 is not a prime number.

2000 and 8750 are written below as the product of their prime factors.

Find the highest common factor (HCF) of 2000 and 8750.

Write 2 × 10¹² as a product of its prime factors.

Write down each of the following.

An even number between 11 and 17.

A square number between 15 and 35.

The cube root of 64.

3 is a factor of 51.

Find a factor of 51 between 10 and 20.

In 2019, comet A and comet B were both seen from Earth. Comet A is seen from Earth every 84 years. Comet B is seen from Earth every 105 years.

Find the next year when both comets will be seen from Earth.

Write 36 as a product of prime factors.

An integer between 70 and 80 is written as the product of its prime factors as 2 × 3 × f.

Find the value of f and the integer.

98 and 147 are written as the product of their prime factors.

98 = 2 × 7² 147 = 3 × 7²

Work out the highest common factor (HCF) of 98 and 147.

In 2019, comet A and comet B were both seen from Earth. Comet A is seen from Earth every 84 years. Comet B is seen from Earth every 105 years.

Find the next year when both comets will be seen from Earth.

The highest common factor (HCF) of two numbers is 14. The lowest common multiple (LCM) of the same two numbers is 210. The two numbers are not 14 and 210.

Find the two numbers.

Ashley has these three number tiles.

Which one of Ashley’s tiles shows a cube number? Write the number on the blank tile on the answer line.

Write down a two-digit prime number that can be made using two of Ashley’s tiles.

Write down the three-digit number closest to 300 that can be made using all three of Ashley’s tiles.

Work out.

4 – 5

2 ×-3

Write down the largest prime factor of 30.

Write down a factor of 18.

Write down a square number between 10 and 20.

Find the two numbers which multiply together to make 40 and add together to make 13.

Some boxes are each in the shape of a cuboid. The base of each box is exactly 35 cm by 45 cm.

The boxes are to be placed on their base, side by side against a wall. If all the shorter sides or if all the longer sides are against the wall, they fit perfectly with no gaps.

Find the shortest possible length of the wall.

Use algebra to prove that an odd number multiplied by a different odd number always gives an

answer that is an odd number.

Complete this statement by writing the missing power in the box.

Blake is asked to write 15 552 000 000 as a product of prime factors in index form. Blake writes

15552000000 = 2⁷_× 5⁶_× 6⁵.

Explain Blake’s mistake.

Write 15 552 000 000 as a product of prime factors in index form.

You are given that 140000 = 2⁵ × 5⁴ × 7.

Find the highest common factor (HCF) of 15 552 000 000 and 140 000.

Write down each of the following.

An odd number.

A square number.

A prime number between 30 and 40.

A multiple of 8.

Li throws two fair four-sided dice, each numbered 1, 2, 3 and 4. Li multiplies together the two numbers that the dice land on to produce a score.

Find the probability that Li’s score is a prime number.

Fountain A squirts water every 24 minutes. Fountain B squirts water every 42 minutes. They squirt water together at 15:19.

Find the next time they squirt water together.

A school sends 60 students from Year 8 and 105 students from Year 9 to a museum.

The school divides these students into groups using the following rules.

• The groups must all be the same size.

• All students in any group must be from the same year.

• There should be as few groups as possible.

Find the size of each group and the total number of groups.

Reece is given this question.

Write 20 as a product of prime factors. Give your answer in index form.

Reece’s answer is 2 × 2 × 5.

Is Reece correct? Explain your answer.

Complete the power of 2.

Work out.

Li throws two fair four-sided dice, each numbered 1, 2, 3 and 4. Li multiplies together the two numbers that the dice land on to produce a score.

Find the probability that Li’s score is a prime number.

Fountain A squirts water every 24 minutes. Fountain B squirts water every 42 minutes. They squirt water together at 15:19.

Find the next time they squirt water together.

A school sends 60 students from Year 8 and 105 students from Year 9 to a museum.

The school divides these students into groups using the following rules.

• The groups must all be the same size.

• All students in any group must be from the same year.

• There should be as few groups as possible.

Find the size of each group and the total number of groups.

Azmi is given this question.

Write 40 as a product of prime factors. Give your answer in index form.

Azmi’s answer is 2 × 2 × 2 × 5.

Is Azmi correct? Explain your answer.

Find the value of x.

Work out.

-1 + 6

7 - - 3

Write down two prime numbers between 10 and 20.

A clock chimes every 20 minutes. A light flashes every 8 minutes. The clock chimes and the light flashes together at 08:00.

How many times between 08:01 and 12:30 will the clock chime and the light flash together? Show your working.

5 is a factor of 20.

Write down another factor of 20.

Write down a multiple of 20.

Write 75 as a product of its prime factors.

A clock chimes every 20 minutes. A light flashes every 8 minutes. The clock chimes and the light flashes together at 08:00.

How many times between 08:01 and 12:30 will the clock chime and the light flash together? Show your working.

Complete this list to show all the factors of 30.

1 2........ 10........ 30

Write down the highest common factor (HCF) of 25 and 30.

A theme park asked 900 people to choose their favourite activity from a list of five. The pictogram shows the results for four of the activities.

How many people chose entertainment?

How many more people chose water rides than family rides?

All 900 people chose one of the five activities.

Complete the pictogram for children’s rides.

Will plays a game at the theme park. There are 20 cards numbered from 1 to 20. Will takes a card at random. He wins if the card he chooses shows a prime number.

Work out the probability that Will wins. Give your answer as a fraction in its simplest form.

A family ticket for the theme park costs £68. If the ticket is bought online it costs 15% less.

How much does it cost to buy a family ticket online?

A number that does not divide exactly by 2........

A number that has no factors except itself and 1........

Write down all the multiples of 4 between 21 and 29.

Write down a common multiple of 4 and 6.

Insert brackets to make this calculation correct.

4 - 1 × 2 = 6

Write 7% as a fraction.

Find the value of

2⁸.

The cube of 3 is added to the square root of 7.

Put a ring around the correct statement.

Two model cars, A and B, are in a race. They start together on the starting line. Assume each car travels at a constant speed.

Car A takes 30 seconds to complete each lap of the track. Car B takes a whole number of seconds to complete each lap of the track. The two cars next cross the starting line together 150 seconds after the start of the race.

Find the four possible times that car B could take to complete one lap.

You may find this information helpful.

At a railway station, trains are either eastbound or westbound. An eastbound train leaves the station every 25 minutes. A westbound train leaves the station every 45 minutes.

An eastbound train and a westbound train both leave the station at 8 am.

When is the next time that two trains leave the station together?

Given that 168 = 2³ × 3 × 7, find the lowest common multiple (LCM) of 168 and 30.

Write down each of the following.

An odd number.

A factor of 25.

A prime number between 20 and 30.

Show that 55 is not a square number.

A bus timetable shows the following information.

• A bus following route T leaves for the train station every 20 minutes.

• A bus following route A leaves for the airport every 18 minutes.

• A bus following route T and a bus following route A both leave at 8.37 am.

When is the next time one of each bus is timetabled to leave at the same time?

Write down one assumption that was necessary to solve this problem.

These are the first five multiples of 15.

15 30 45 60 75

Write down the first five multiples of 30.

(a)........,........,........,........,........

Write down the lowest common multiple (LCM) of 15 and 30.

A bus timetable shows the following information.

• A bus following route T leaves for the train station every 20 minutes.

• A bus following route A leaves for the airport every 18 minutes.

• A bus following route T and a bus following route A both leave at 8.37 am.

When is the next time one of each bus is timetabled to leave at the same time?

Write down one assumption that was necessary to solve this problem.

You are given that

Find the lowest common multiple (LCM) of 270 and 177 147. Give your answer using power notation and as an ordinary number.

(a)(i) using power notation........

as an ordinary number........

Write 177 147 000 000 as a product of its prime factors.

3ⁿ = 177147 × 9⁵.

Find the value of n.

Prove that the mean of any four consecutive even integers is an integer.

Write down each of the following.

An even number.

A factor of 25.

A prime number between 10 and 20.

A cube number.

Find the highest common factor (HCF) of 35 and 91.

Doctor Jones starts an appointment every 20 minutes. Doctor Warholm starts an appointment every 35 minutes.

The first appointment for both doctors starts at 8.30 am.

What is the next time that they have an appointment start at the same time?

Geoff has two fair spinners.

He spins both spinners and multiplies the numbers on each spinner.

Complete the table.

Geoff wants to work out the probability that the outcome of the multiplication is an even number or a prime number. Here is his working.

The probability the outcome is an even number is The probability the outcome is a prime number is The probability the outcome is an even number or a prime number is =

Geoff is wrong. Explain his error and give the correct answer.

Two numbers, P and Q, are written as products of their prime factors.

Find the lowest common multiple (LCM) of P and Q.

The number C is written as the product of its prime factors.

Work out P ÷ C, leaving your answer as a product of powers of prime numbers.

Write 450 as a product of its prime factors.

Find the highest common factor (HCF) of 270 and 450.

Geoff has two fair spinners.

He spins both spinners and multiplies the numbers on each spinner.

Complete the table.

Geoff wants to work out the probability that the outcome of the multiplication is an even number or a prime number. Here is his working.

The probability the outcome is an even number is The probability the outcome is a prime number is The probability the outcome is an even number or a prime number is =

Geoff is wrong. Explain his error and give the correct answer.

Here is a list of numbers.

2 8 5 12 6

From this list, write down

the odd number,

the cube number.

Using the same list of numbers, work out

the median,

the range.

Show that the highest common factor (HCF) of 18 and 63 is 9.

Find the lowest common multiple (LCM) of 18 and 63.

Write down a multiple of 6 between 10 and 20.

Write down two factors of 30 that are prime numbers.

Write 120 as a product of its prime factors.

The lowest common multiple (LCM) of x and 120 is 360.

Find the smallest possible value of x.

Two numbers, A and B, are written as a product of prime factors.

A = 2⁴ × 3² × 7² B = 2³ × 3 × 5 × 7

Find the highest common factor (HCF) of A and B.

Write down

a multiple of 13,

a prime number between 40 and 50.

Find the lowest common multiple (LCM) of 16 and 28.

Andrew is thinking of a number.

• It is between 1 and 150.

• It is one more than a square number.

• It is three less than a cube number.

• It is not a prime number.

What is Andrew’s number? You must show all your reasoning.

n is an integer.

Explain why 2 n + 1 is an odd number.

Write down an algebraic expression for the next odd number after 2 n + 1.

Use algebra to show that the sum of two consecutive odd numbers will always be a

multiple of 4.

Andrew is thinking of a number.

• It is between 1 and 150.

• It is one more than a square number.

• It is three less than a cube number.

• It is not a prime number.

What is Andrew’s number? You must show all your reasoning.

Prove that the difference between two consecutive square numbers is always odd.

Carla runs every 3 days. She swims every Thursday. On Thursday 9 November, Carla both runs and swims.

What will be the next date on which she both runs and swims?

Here is a list of numbers.

11 27 81 21 41 42 23 39 45

From this list, write down

the even number,

the square number,

all the prime numbers.

Write 490 as the product of its prime factors.

Buses to Ayton leave the station every 25 minutes. Buses to Bleeford leave the station every 40 minutes. Buses to both places leave at 9am.

What is the next time buses to Ayton and Bleeford leave the station together?

Two numbers have these properties.

• Both numbers are greater than 6.

• Their highest common factor (HCF) is 6.

• Their lowest common multiple (LCM) is 60.

Find the two numbers.

Write 504 as the product of its prime factors.

Find the lowest common multiple (LCM) of 180 and 504.

n is an integer.

Explain why 2 n + 1 is an odd number.

Prove that the difference between the squares of two consecutive odd numbers is a multiple of 8.