Question

Prove or disprove each of the following statements by exhaustive checking. a. There is a prime number between 45 and 54. b. The product of any two of the four numbers 2, 3, 4, and 5 is even. c. Every odd integer between 2 and 26 is either prime or the product of two primes. d. If d|ab, then d|a or d|b. e. If m and n are integers, then (3m + 2)(3n + 2) has the form (3k + 2) for some integer k. f. The sum of two prime numbers is a prime number. g. The product of two prime numbers is odd. h. There is no prime number between 293 and 3

Answer 1

a)Correct ,b)Incorrect ,c)Correct ,d) Incorrect ,e) Incorrect ,fIncorrect ,g)Incorrect ,h)Correct

Explanation:

a)

47 is the prime number between 45 and 54 so this statement is true,

b)

The product of 3 and 5 is 15 which is an odd number so the statement that the product of any two of 2,3,4,5 is even is incorrect.

c)

list of odd integers between 2 and 26 = 3,5,7,9,11,13,15,17,19,21,23,25

where 3,5,7,11,13,17,19,23, are all prime numbers while remaining number 9,15,21,25 are all product of two prime number i.e, 9=3*3, 15=3*5, 21=7*3, 25=5*5.

So, the statement that every odd integer between 2 and 26 is either prime of product of two prime numbers is true.

d)

lets consider d=6, a=3, b=4

Since

6I12, but 6∦3 or 6∦4 thus if dIab, then it does not imply that dIa or dIb

So, the statement that if dIab, then dIa or dIb is incorrect.

e)

lets consider m=0 and n=1

then we get 3m+2 = 3(0)+2=2

3n+2 = 3(1) +2= 5

So, their product is 2(5) = 10 but the given equation cannot be written as 3k+2=10 as it would not yield integer solution. Therefore the given statement is incorrect.

f)

3 and 5 are both prime numbers but their sum 8 is not a prime number so the statement that sum os two prime nubers is also a prime number is incorrect.

g)

2 and 3 are both prime numbers but their product 6 is an even number so the statement that product of two prime numbers is a prime number is incorrect.

h)

There is no prime number between 293 and 307. After 293 the next prime number is 307, so the statement that there is no prime number between 293 and 300 (307) is correct.

Answer 2

Explanation:

We need to prove or disprove the above statements by exhaustive checking.

In exhaustive checking, when a statement asserts that each of a finite number of things has a certain property, then we might be able to able to prove the statement by checking that each thing has the stated property.

a. There is a prime number between 45 and 54.

* Between 45 and 54, 47 is a prime number.

Hence, the statement is TRUE.

b. The product of any two of the four numbers 2, 3, 4, and 5 is even.

2×3 = 6

2×4 = 8

2×5 = 10

3×4 = 12

3×5 = 15

4×5 = 20

The products: 6, 8, 10, 12, and 20 are even, BUT 15 is odd. Hence,

the statement is FALSE.

c. Every odd integer between 2 and 26 is either prime or the product of two primes.

Odd integers between 2 and 26 are:

3, 5, 7, 11, 13, 15, 17, 19, 21, 23, and 25.

Prime numbers are: 3, 5, 7, 11, 13, 17, 19, 23.

Product of Primes: 15 = 3×5, 21 = 7×3, 25 = 5×5

Hence, this statement is TRUE.

d. If d/ab, then d/a or d/b.

If d divides ab, then d divides a or d divides b.

d/ab = d/a × 1/b or d/b × 1/a.

120 divides 5×2

120 divides 10

=> 120 divides (5×2)

=> 120 divides 5 or 120 divides 2.

Hence, the statement is TRUE.

e. If m and n are integers, then (3m + 2)(3n + 2) has the form (3k + 2) for some integer k.

If m = 2, and n = 3

Then

(3m + 2)(3n + 2) = (3×2 + 2)(3×3 + 3)

= (6+2)(9+3) = 8×11 = 88

Notice that in the form (3k + 2),

88 = 3(86/3) + 2

But 86/3 is not an integer.

So, the statement is FALSE

f. The sum of two prime numbers is a prime number.

11, 7, and 5 are all prime numbers,

11+7 = 19 is prime,

but 11+5 = 16 is not a prime. This statement is either TRUE or FALSE.

g. The product of two prime numbers is odd.

2, 3, and 5 are all prime numbers,

3×5 = 15 is an odd number,

but 2×5 = 10 is not an odd number. This statement is either TRUE or FALSE.

h. There is no prime number between 293 and 3

5 is a prime number. This statement is FALSE.