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mathematics UK Year 10 Mathematics (KS4)
 
Solved problems for the chapter:
 

Types of number

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Multiples and expressions for KS4
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The greatest common divisor and the least common multiple
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Detailed solution steps are available at:
 
https://en.openprof.com/wb/chapter:types_of_number/5300/


Example 1.

For arbitrary natural numbers m and n, denote by the greatest common divisor of these two numbers and by their least common multiple.


  • Write the numbers 45, 48, and 60 as products of their prime factors.


  • Calculate

   
Example 2.

Write the first five multiples of 5 and 9.

    
3.

Given the number 1008:


  • express it as a product of prime numbers;

  • express its square root in surd form.

    
4.

Find the digits a and b for which the number 27a874b is divisible by the number 2.

    
5.

We have the given number 4213a. For which number a will a given number be divisible by 6?

    
6.

State whether each of the following statements are correct or incorrect?


  • - 2 is a natural number.


  • is a real number.


  • is an irrational number.


  • is a rational number.

    
7.

Evaluate the expression

find the multiple of the number 2 for each natural number x?

    
8.

Find the possible values of the digit a for which the number 37a586 is divisible by 3. Is the given number also divisible by 8?

    

9.

Find the pairs of natural numbers {a, b} for which and .

    
10.

Write down the additive inverse of the given numbers below:


  • ,

  • ,

  • ,

  • ,

  • .

    
11.

People aged 15 to 75 in London were asked which social network they use most often. The data is shown in the chart. See the chart and answer the questions below.




  • Which social network is the most used.

  • Which social network is the least frequently used.

  • Does Twitter use more respondents than Instagram?

  • Which social networks do more than 10% of respondents use?

  • How many social networks are represented in the chart?

  • Check if more than half of the respondents answered that they use Youtube or Twitter?

    
12.

Obtain the absolute values of rational numbers below:













    
13.

For a given number find the values for b so that the number is divisible by 3 and not by 2.

    
14.

Insert or where appropriate.


  •  


  •  


  •  


  •  


   
15.
Find the values for the variable by giving the number



divisible by 11.

    
16.

Show that the value of the given expression:




is a multiple of 6.

    
17.

Write down:

  • five rational numbers that are also positive integers,

  • five rational numbers that do not belong to the set of natural numbers, but belong to the set of integers,

  • five rational numbers that do not belong to the set of integers.

    
18.

We have two consecutive even numbers and , which is considered to be . Determine whether the difference of the squares of these two numbers is divisible by .

    

19.

The sum of three consecutive natural numbers is an odd number. By what common divisor are all their products divisible?

    
20.

Given the number 12350214. Read the statements below carefully. If the statement is correct, circle YES, if it is incorrect, NO.


  • The number is divisible by 1. YES NO

  • The number is divisible by 2. YES NO

  • The number is divisible by 3. YES NO

  • The number is divisible by 4. YES NO

  • The number is divisible by 5. YES NO

  • The number is divisible by 6. YES NO

  • The number is divisible by 9. YES NO

  • The number is divisible by 10. YES NO

    
21.

A six-digit number is given 2345a1. Determine the digits a to get the numbers that are divisible by 3.

    
22.

The fourteen-digit number is given. Write down all the options for the digit a for which is the given number is:


divisible by 2:  

divisible by 3:  

divisible by 4:  

divisible by 5:  

divisible by 6:  

divisible by 9:  

divisible by 10:      
23.

Show that the value of the expression:




is divisible by 4.

    
24.

Show that the given expression:




is a multiple of 21.

    
25.

Which of the numbers 2, 3, 4, 5, 6, 8 and 9 is the number 102030405060 divisible by? Mark the correct answers in the table.


The number 102030405060 is divisible by:


2DivisibleNot divisible 
3DivisibleNot divisible 
4DivisibleNot divisible 
5DivisibleNot divisible 
6DivisibleNot divisible 
8DivisibleNot divisible 
9DivisibleNot divisible 
    
26.

A number n divides a dividend to give a quotient of 7 and remainder 1. If also divides the dividend to give a quotient 5 and remainder of 3. Find n and the dividend.

    
27.

Find a number that divides the difference of the squares of two consecutive odd numbers, where the two consecutive odd numbers are and .

    
28.

If and are divisible by . Prove that is also divisible by .

    

29.

Prove that is divisible by 9 if is divisible by 6 and is a multiple of 5.

    
30.

Prove that:


If n is not divisible by 3, then its square reduced by 1 is divisible by 3.

    
31.

If is a perfect square, find the minimum even number which satisfies the expression.

    
32.

Prove that if n is a number, then is divisible by 4.