In a mathematical argument, how are three consecutive integers usually denoted algebraically?
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In a mathematical argument, how are three consecutive integers usually denoted algebraically?
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Prove that the sum of two odd numbers is even.
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Explain why for all real values of .
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Prove that the product of two even numbers is a multiple of 4.
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Use a counter-example to show that .
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Prove by exhausting all possible factors that 11 is a prime number.
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Prove that for all real values of .
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Show that 0.6 can be written in the from , where and are integers.
What does this tell you about the type of number 0.6 is?
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Prove that the sum of any three consecutive integers is a multiple of 3.
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Prove that for all values of .
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Prove that the square of an even number is a multiple of 4.
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The set of numbers S is defined as all positive integers less than 5.
Prove by exhaustion that the cube of all values in S are less than 100.
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Use a counter-example to prove that the difference between any two square numbers is not always odd.
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Express 18 as a product of its prime factors.
Write down all prime numbers between 1 and 13.
By dividing 13 by each of the prime numbers found in part (b), prove that 13 is a prime number.
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Factorise .
Hence show that .
Given that n is even, write down whether are odd or even.
Hence deduce whether is odd or even. Justify your answer.
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By writing it as a fraction in its lowest terms, show that 0.35 is a rational number.
Two rational numbers, and are such that and where are integers with no common factors and .
Find an expression for .
Deduce whether or not the product is rational or irrational.
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Prove that a triangle with side lengths of 8 cm, 6 cm and 10 cm must contain a right-angle. You may use the diagram below to help.
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A standard chess board has 64, 1x1- sized squares.
It also has 1, 8x8 - sized square.
How many 2x2 - sized squares are there on a standard chess board?
Write down the number of 3x3 - sized and 4x4 - sized squares there are on a standard chess board.
Hence show that there are 204 squares in total on a standard chess board.
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Prove that the sum of any three consecutive even numbers is a multiple of 6.
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Prove that for all values of , where .
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Prove that the square of an odd number is always odd.
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The set of numbers S is defined as all positive integers greater than 5 and less than 10.
Prove by exhaustion that the square of all values in S differ from a multiple of 5 by 1.
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Use a counter-example to prove that not all integers of the form , where is an integer, are prime.
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By considering all possible prime factors of 17, prove it is a prime number.
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Fully factorise .
Prove that, if is odd, is odd and that if is even, is even.
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Two rational numbers, a and b are such that and , where are integers with no common factors and .
Find expressions for and .
Deduce whether or not and are rational or irrational.
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Prove that the exterior angle in any triangle is equal to the sum of the two opposite interior angles. You may use the diagram below to help
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A standard chess board has 64, 1x1 - sized squares.
It also has 1, 8x8 - sized square.
How many 2x2 - sized and 3x3 - sized squares are there on a standard chess board?
Hence show that there are 204 squares in total on a standard chess board.
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Prove that the sum of any three consecutive even numbers is always a multiple of 2, but not always a multiple of 4.
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Prove that for all values of , where .
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Prove that the (positive) difference between an integer and its cube is the product of three consecutive integers.
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The elements, , of a set of numbers, S, are defined .
Prove that every element of S can be written in the form where .
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Give an example to show when the following statement is both true and false.
The square of a positive integer is always greater than doubling it.
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Prove that 23 is a prime number.
Briefly explain why only prime factors need to be tested for, in order to prove a number is prime.
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Prove that, if is negative, is positive.
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Prove that the sum of two rational numbers is rational.
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Prove the angle at the circumference in a semi-circle is a right angle.
You may use the diagram below to help.
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A standard chess board has 64 1x1 - sized squares.
It also has 1 8x8 - sized square.
Prove that there are 204 squares on a standard chess board.
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