Investigations (Cambridge (CIE) IGCSE International Maths: Core): Exam Questions

The following method is used to prove that a number, , is a prime number:

• Find

• List all the prime numbers less than or equal to

• Check if is divisible by each of these prime numbers

The prime numbers 2, 3 and 5 have well-known divisibility tests that are often quicker to do than using a calculator (shown below). The divisibility test for 11 (from question 1) can now be added to this list.

This leaves divisibility by 7, 13, 17 and 19 to be done using a calculator.

For example, to prove that 313 is prime, check if it is divisible by all prime numbers less than or equal to

These are 2, 3, 5, 7, 11, 13 and 17:

As 313 is not divisible by any prime number less than or equal to , the conclusion is that 313 is a prime number.

The row for divisibility by 19 was not needed (since ).

Use the same table above to prove that the number 191 is a prime number.

Leave blank any rows that are not needed.

Digits can only take the integer values 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. A digit cannot be negative.

The five-digit number "" is divisible by 11.

Use the test in question 1 to find the value of the digit .

The five-digit number"" is divisible by 11 and has an alternating-digit sum of zero.

Use the table below to list all the possible values that and can take.

The first row has been done for you.

If the five-digit number "" described in part (b) is also divisible by 3, find the only possible values of and .

Explain your answer.

A four-digit number in the form "" is always divisible by 11. This is because the divisibility test in question 1 gives

Show that a four-digit number in the form "" is always divisible by 11.

A student believes that four-digit numbers in the form "" where are always divisible by 11.

Use an example of a number in the form "" and the divisibility test in question 1 to prove that the claim is not true.

Determine which out of the following six-digit numbers are always divisible by 11.

You may assume that .

The first row has been done for you.

The number "" is an example of an eight-digit number that is always divisible by 11.

Using four digits of and four digits of , find two more examples of eight-digit numbers that are always divisible by 11.

Label the first digit as .

A number is called palindromic if it is equal to itself when the digits are written in reverse order.

For example, the numbers 33, 181, 9999, 12321 and 579975 are all palindromic.

Explain why the number 812128 is not palindromic.

All palindromic numbers with an even number of digits are divisible by 11.

The table below can be used to prove this result for four, six and eight-digit palindromic numbers.

Complete the table.

The first row has been done for you.

Palindromic numbers with an odd number of digits are not always divisible by 11.

Prove this statement using the five digit palindromic numbers 14641 and 13831.

It is possible to form palindromic numbers by squaring multiples of .

Determine which, out of , and , give palindromic numbers.

Squaring larger multiples of 11 can produce larger palindromic numbers.

Find a six-digit palindromic number that is the square of a multiple of 11.

You may use the table below to help.

It is possible to form palindromic numbers by adding together the squares of two different multiples of .

For example:

Find two more examples of palindromic numbers that are formed by adding together the squares of two different multiples of .

You may use the grid below to help.

A student says that the square of any positive multiple of 13 is never a palindromic number.

Investigate this claim and decide whether, or not, the student is correct.

Show your working clearly.

A palindromic prime is a prime number that is also palindromic.

A two-digit palindromic prime is 11.

Use the information in question 2 to write down two different three-digit palindromic primes.

Show that the number 111 is not a palindromic prime.

Explain clearly why palindromic primes, other than 11, cannot have an even number of digits.

Use the information in question 5(b) to help.

A second test to determine whether or not a number is divisible by 11 is as follows:

• Sum all the digits in the odd positions (1^st digit + 3^rd digit + 5^th digit + ...)

• Sum all the digits in the even positions (2^nd digit + 4^th digit + 6^th digit +...)

• Subtract the second sum from the first sum

• If the result is a multiple of 11, the original number is divisible by 11

An eight-digit number is given by "".

Use algebra to prove that, for eight-digit numbers, this test is mathematically the same as the test in question 1.