Questions 1-9 : Find which of the following pairs of numbers are relatively prime.
Question 1 :
4 and 12
Question 2 :
7 and 43
Question 3 :
5 and 3
Question 4 :
8 and 17
Question 5 :
8 and 15
Question 6 :
14 and 21
Question 7 :
2 and 4
Question 8 :
1 and 2
Question 9 :
8, 15 and 49
Question 10 :
The LCM of two relatively prime numbers is 5005. If one of the numbers is 65, then find the other number.
Question 11 :
If the product of two relatively prime numbers is 117, find the LCM of the numbers.
Question 12 :
If two numbers are relatively prime, then, which is of the following must be true about the two numbers? Explain.
(A) Both the numbers must be prime
(B) One must be prime and other must be composite
(C) Both the numbers must be composite
(D) Both of them can be any numbers
1. Answer :
4 and 12 have the common divisors other than 1.
They are 2 and 4.
So, 4 and 12 are not relatively prime.
2. Answer :
7 and 43 have no common divisor other than 1.
So, 7 and 43 are relatively prime.
3. Answer :
5 and 3 have no common divisor other than 1.
So, 5 and 3 are relatively prime.
4. Answer :
8 and 17 have no common divisor other than 1.
So, 8 and 17 are relatively prime.
5. Answer :
8 and 15 have no common divisor other than 1.
So, 8 and 15 are relatively prime.
6. Answer :
14 and 21 have common divisor other than 1.
That is 7.
So, 14 and 21 are not relatively prime.
7. Answer :
2 and 4 have common divisor other than 1.
That is 2.
So, 2 and 4 are not relatively prime.
8. Answer :
1 and 2 have no common divisor other than 1.
So, 1 and 2 are relatively prime.
9. Answer :
Resolve 8, 15 and 49 into their prime factors.
8 = 2 x 2 x 2
15 = 3 x 5
49 = 7 x 7
There is no common factor or divisor for the numbers 8, 15 and 49.
So, the numbers 8, 15 and 49 are relatively prime.
10. Answer :
Let x be the other number.
We know that,
Product of the two numbers = LCM x HCF
As the HCF of two relatively prime numbers is 1,
65(x) = 5005 x 1
65x = 5005
Divide each side by 65.
x = 77
The other number is 77.
11. Answer :
Because the two numbers are relatively prime, their HCF is 1.
We know that,
LCM x HCF = Product of the two numbers
Substitute.
LCM x 1 = 117
LCM = 117
12. Answer :
The correct answer choice is (D).
That is, both of them can be any numbers.
Consider the following pairs of relatively prime numbers.
(2, 3) ----> both of them are prime
(4, 5) ----> one is composite and other one is prime
(8, 15) ----> both of them are composite
From the above examples, it is clear that if two two numbers are relatively prime, they can be any numbers.
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