Find the prime factorisation of the following numbers
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Find the prime factorisation of the following numbers
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State whether the following are rational or irrational quantities.
For those that are rational, write them in the form , where are integers and is in its simplest terms.
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Prove by contradiction that the sum of two consecutive integers is odd.
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Prove by contradiction that the product of two odd numbers is odd.
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Prove by contradiction that if is even, then must be even.
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Prove by contradiction that there is an infinite number of multiples of 10.
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Prove by contradiction that if is odd, then must be odd.
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When a number is rational, it can be written in the form .
Two numbers can be written in the form and such that meet the necessary conditions so that the two numbers are rational.
Prove that the product of these numbers is also rational.
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Prove by contradiction that there are an infinite number of even numbers.
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A student is attempting to answer the following exam question:
“Prove by contradiction that is an irrational number. You may use without proof the fact that if a number is even, then n must also be even.”
The student’s proof proceeds as follows:
Line 1: |
Assume is a rational number. Therefore, it can be written in the form , where are integers with , and where and may be assumed to have no common factors. |
Line 2: | Squaring both sides: |
Line 3: | |
Line 4: | |
Line 5: | Therefore , for some integer |
Line 6: | Then, |
Line 7: | |
Line 8: | |
Line 9: | So is even and therefore is also even. |
Line 10: | It has been shown that both and are even, so they share a common factor of 2. |
Line 11: | This is a contradiction of the assumption that and have no common factors. |
Line 12: | Therefore, is irrational. |
There is an error within the first three lines of the proof.
State what the error is and write the correct line down.
Line 4 of the proof is missing.
Write down the missing line of the proof.
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A composite number can be written uniquely as the product of its prime factors.i.e., any composite number N can be written uniquely as , where are the prime factors of N.
Show that a composite number N may be written in the form , where is an integer and is a prime factor of N.
By expressing in terms of the prime factors of N, be sure to explain why must be an integer.
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Prove by contradiction that a triangle cannot have more than one obtuse angle.
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Prove by contradiction that if is odd, then must be odd.
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Prove that the product of two rational numbers is rational.
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Prove by contradiction that there are an infinite number of powers of 2.
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Prove by contradiction that is an irrational number. You may use without proof the fact that if is a multiple of 11, then is a multiple of 11.
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Below is a proof by contradiction that there is no largest multiple of 7.
Line 1: | Assume there is a number, S, say, that is the largest multiple of 7. |
Line 2: | |
Line 3: | Consider the number . |
Line 4: | |
Line 5: | |
Line 6: | So is a multiple of 7. |
Line 7: | This is a contradiction to the assumption that S is the largest multiple of 7. |
Line 8: | Therefore, there is no largest multiple of 7. |
The proof contains two omissions in its argument.
Identify both omissions and correct them.
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If a positive integer greater than 1 is not a prime number, then it is called a composite number. Prove by contradiction that any composite integer N has a prime factor less than or equal to .
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Prove by contradiction that if is odd, where is a positive integer, then must be odd.
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Prove that the difference between two rational numbers is rational.
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Prove by contradiction that there are an infinite number of prime numbers.
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Prove by contradiction that , where is a prime number, is an irrational number. You may use without proof the fact that any positive integer may be written uniquely as a product of its prime factors.
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Below is a proof by contradiction that is irrational.
Line 1: |
Assume is a rational number. Therefore it can be written in the form , where a and b are integers, and . As , we may assume as well that . |
Line 2: | |
Line 3: | |
Line 4: | |
Line 5: | No power of 2 (all even) is equal to a power of 7 (all odd). |
Line 6: | unless but this is a contradiction of the original assumption. |
Line 7: | is irrational |
The proof contains one mathematical error and one logical error.
Identify both errors and correct them.
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Prove by contradiction that the solutions to the equation cannot be written in the form where and are both odd integers.
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Prove by contradiction that, if are rational numbers and is a positive non-square integer, then
implies that and . You may use without proof the fact that for any positive non-square integer , is irrational.
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