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Discrete Mathematics and its Applications

Kenneth H. Rosen

$Math Studyset Vaia Explanations$ Math

7th Edition · 808 pages

ISBN: 9780073383095

Chapter 4: Q19SE (page 307)

Show that every integer greater than 11 is the sum of two composite integers.

Short Answer

Expert verified

If n > 11, it can be written as a sum of two composite numbers.

Step by step solution

Step 1

Note that the only triplet of the form k,k+2,k+4which are all primes is when k = 3.

This can be see as follows.

If does not divide kand k + 2, then it should divide k + 1which implies that 3divides k + 4.

Now suppose that a number ncannot be written as a sum of two composite numbers.

Step 2

This implies that n - 4,n - 6 and n - 8 are all primes (since 4,6,8are composite).

This implies from above that n - 8 = 3which implies that n = 11 .

Thus if n > 11, it can be written as a sum of two composite numbers.

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Most popular questions from this chapter

The value of the Euler $ϕ$ -function at the positive integer is defined to be the number of positive integers less than or equal to that are relatively prime to. [Note: $ϕ$ is the Greek letter phi.]

Find these values of the Euler $ϕ$ -function.

a)role="math" localid="1668504243797" $ϕ (4)$ b)role="math" localid="1668504251452" $ϕ (10)$ c)role="math" localid="1668504258881" $ϕ (13)$

Determine whether the integers in each of these sets are Pairwise relatively prime.

a) 21, 34, 55 b) 14, 17, 85

c) 25, 41, 49, 64 d) 17, 18, 19, 23

How many divisions are required to find gcd(21,34) using the Euclidean algorithm?

What are the greatest common divisors of these pairs of integers?

a) $2^{2} \cdot 3^{3} \cdot 5^{5}, 2^{5} \cdot 3^{3} \cdot 5^{2}$

b) $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13, 2^{11} \cdot 3^{9} \cdot 11 \cdot 17^{14}$

c) 17, $17^{17}$

d) $2^{2} \cdot 7, 5^{3} \cdot 13$

e) 0, 5

f) $2 \cdot 3 \cdot 5 \cdot 7, 2 \cdot 3 \cdot 5 \cdot 7$

Show that if $2^{m} + 1$ is an odd prime, then $m = 2^{n}$ for some nonnegative integer $n$ . [Hint: First show that the polynomial identity $x^{m} + 1 = (x^{k} + 1) (x^{k (t - 1)} - x^{k (t - 2)} + \dots - x^{k} + 1)$ holds, where $m = k t$ and $t$ is odd.]

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Show that every integer greater than 11 is the sum of two composite integers.

Short Answer

Step by step solution

Step 1

Step 2

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Most popular questions from this chapter

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