Chapter 1 - Real Numbers, RD Sharma Solutions - (Part - 12) | RD Sharma Solutions for Class 10 Mathematics PDF Download

Page No 1.62

Q.16. If p and q are two prime number, then what is their HCF?
Ans. It is given that p and q are two prime numbers; we have to find their HCF.
We know that the factors of any prime number are 1 and the prime number itself.
For example, let p = 2 and q = 3
Thus, the factors are as follows
p = 2 x 1
And
q = 3 x 1
Now, the HCF of 2 and 3 is 1.
Thus the HCF of p and q is 1.

Q.17. If p and q are two prime number, then what is their LCM?
Ans. It is given that p and q are two prime numbers; we have to find their LCM.
We know that the factors of any prime number are 1 and the prime number itself.
For example, let p = 2 and q = 3
Thus, the factors are as follows
p = 2 x 1
And
q = 3 x 1
Now, the LCM of 2 and 3 is 2 x 3 = 6.
Thus the HCF of p and q is p x q.

Q.18. What is the total number of factors of a prime number?
Ans. We know that the factors of any prime number are 1 and the prime number itself.
For example, let p = 2 
Thus, the factors are as follows
p = 2 x 1
Hence, the total number of factors of a prime number is 2.

Q.19. What is a composite number?
Ans. A composite number is a positive integer which has a divisor other than one or itself.
In other words a composite number is any positive integer greater than one that is not a prime number.

Q.20. What is the HCF of the smallest composite number and the smallest prime number?
Ans. The smallest composite number is 4
The smallest prime number is 2
Thus, the HCF of and is 2.

Q.21. HCF of two numbers is always a factor of their LCM (True/False).
Ans. HCF of two numbers is always a factor of their LCM
True
Reason:
The HCF is a factor of both the numbers which are factors of their LCM.
Thus the HCF is also a factor of the LCM of the two numbers.

Q.22. π is an irrational number (True/False).
Ans. Here π is an irrational number
True
Reason:
Rational number is one that can be expressed as the fraction of two integers.
Rational numbers converted into decimal notation always repeat themselves somewhere in their digits.
For example, 3 is a rational number as it can be written as 3/1 and in decimal notation it is expressed with an infinite amount of zeros to the right of the decimal point. 1/7 is also a rational number. Its decimal notation is 0.142857142857…, a repetition of six digits.
However √2 cannot be written as the fraction of two integers and is therefore irrational. 
Now,
π = 3.14159265358979323846264338327950288419716939937510...
Thus, it is irrational.

Q.23. The sum of two prime number is always a prime number (True/ False).
Ans. The sum of two prime numbers is always a prime number.
False
Reason:
Let us prove the above by taking an example.
Let the two given prime numbers be 2 and 7.
Thus, their sum, i.e; 9 is not a prime number.
Hence the above statement is false

Q.24. The product of any three consecutive natural number is divisible by 6 (True/False).
Ans. The product of any three natural numbers is divisible by 6.
True
Reason:
Let the three consecutive natural numbers be 1,2 and 3.
Their product is 6, which is divisible by 6
Let the other set of three consecutive natural numbers be 3, 4 and 5.
Their product is 60, which is divisible by 6

Q.25. Every even integer is of the form 2m, where m is an integer (True/False).
Ans. Every even integer is of the form 2m, where m is an integer (True/False)
True
Reason:
Let the various values of m as -1, 0 and 9.
Thus, the values for 2m become -2, 0 and 18 respectively.

Q.26. Every odd integer is of the form 2m − 1, where m is an integer (True/False).
Ans. Every odd integer is of the form 2m - 1, where m is an integer (True/False)
True
Reason:
Let the various values of m as -1, 0 and 9.
Thus, the values for 2m - 1 become -3, -1 and 17 respectively.
These are odd integers.

Q.27. The product of two irrational numbers is an irrational number (True/False).
Ans. The product of two irrational numbers is an irrational number (True/False)
False
Reason:
Let us assume the two irrational numbers be √2 and √3
Sometimes, it is and sometimes it isn't.
√2 And √3 are both irrational as their product is √6
Now √2 and √8 are both irrational but their product, √16 is rational (in fact, it
equals 4)

Q.28. The sum of two irrational number is an irrational number (True/False).
Ans. 
The sum of two irrational numbers is an irrational number (True/False)
False
Reason:
However, √2 is not rational because there is no fraction, no ratio of integers that will equal √2. It calculates to be a decimal that never repeats and never ends. The same can be said for √3. Also, there is no way to write √2 + √3 as a fraction. In fact, the representation is already in its simplest form.
To get two irrational numbers to add up to a rational number, you need to add irrational numbers such as 1 + √2 and 1 - √2. In this case, the irrational portions just happen to cancel out leaving: 1 + √2 + 1 - √2 = 2 which is a rational number (i.e. 2/1).

Q.29. For what value of n, 2ⁿ ✕ 5ⁿ ends in 5.
Ans. We need to find the value of n, for which 2ⁿ ✕ 5ⁿ ends in 5.
Clearly,
2ⁿ ✕ 5ⁿ = (2 x 5)ⁿ
= 10ⁿ
Also, all the values of n will make 10ⁿ end in 0.
Thus, there is no value of n for which 2ⁿ ✕ 5ⁿ ends in 5.

Q.30. If a and b are relatively prime numbers, then what is their HCF?
Ans. It is given that a and b are two relatively prime numbers; we have to find their HCF.
We know that two numbers are relatively prime if they don’t have any common divisor.
Also, the factors of any prime number are 1 and the prime number itself.

For example, let a = 7 and b = 20
Thus, the factors are as follows
a = 7 × 1
And
b = 2² × 5 × 1
Now, the HCF of 7 and 20 is 1.
Thus the HCF of a and b is 1.

Q.31. If a and b are relatively prime numbers, then what is their LCM?
Ans. It is given that a and b are two relatively prime numbers; we have to find their LCM.
We know that two numbers are relatively prime if they don’t have any common divisor.
Also, the factors of any prime number are 1 and the prime number itself.
For example, let a = 7 and b = 20
Thus, the factors are as follows
a = 7 × 1
And
b = 2² × 5 × 1
Now, the LCM of 7 and 20 is 140.
Thus the HCF of a and b is ab.

Q.32. Two numbers have 12 as their HCF and 350 as their LCM (True/False).
Ans. Two numbers have 12 as their HCF and 350 as their LCM (True/False).
False.
Reason:
We know that HCF should divide LCM.
But, the HCF 12 does not divide the LCM 350.

Page No 1.63

Q.33. Find after how many places of decimal the decimal form of the number 27/2³.5⁴.3² will terminate.
Ans. 
27/2³.5⁴.3² = 3^3/2³.5⁴.3²
= 3/2³.5⁴
= 3/8 × 625
= 3/5000
= 0.003/5
= 0.0006
Hence, after four places of decimal the decimal form of the number 27/2³.5⁴.3² will terminate.

Q.34. Express 429 as the product of its prime factors.
Ans. The prime factorisation of 429 is
429 = 3 × 11 × 13 

Q.35. Two positive integers a and b can be written as a = x³y² and b = xy³, where x, y are prime numbers. Find LCM (a, b).
Ans. It is given that, a = x³y² and b = xy³, where x, y are prime numbers.
LCM (a, b) = LCM (x³y²,  xy³)
= The highest of indices of x and y
=  x³y³
Hence, LCM (a, b) is  x³y³. 

Q.36. If HCF (336, 54) = 6, find LCM (336, 54).
Ans. we know that,
Product of two numbers = LCM × HCF
⇒ 336 × 54 = LCM × 6
⇒ LCM × 6 = 336 × 54
⇒ LCM = 336 × 54/6
⇒ LCM = 336 × 9
⇒ LCM = 3024
Hence, LCM (336, 54) is 3024.