Important Questions Class 10 Maths Chapter 1 Real Numbers

In mathematics, the exploration of real numbers is an exciting exercise. Important Questions Chapter 1 – Real Numbers take you on a ride where you encounter the combination of both rational numbers such as integers or fractions and irrational numbers such as root numbers. They can be both positive and negative. All arithmetic calculations can be done with them and can be represented over a number line. Real numbers are denoted as ‘R’.

While evaluating important questions in class 10 maths Real Numbers Chapter 1, students are required to pay attention while dealing with the following topics:

• Highest Common Factor(HCF)  and Least Common Multiple(LCM) 
• Decimal representation of irrational numbers (eg 3.4, 48.2)
• Commutative, associative, and distributive properties of addition and multiplication
• Euclid’s division algorithm (Euclid’s division lemma)
• Fundamental theorem of arithmetic 

Important Questions Chapter 1 – Real Numbers

Q1. If the HCF of 408 and 1032 is expressible in the form 1032 × 2 + 408 × p, then find the value of p.

Solution:

HCF of 408 and 1032 is 24.

1032 × 2 + 408 × (p) = 24

408p = 24 – 2064

p = -5

Q2. Find the largest number which divides 70 and 125 leaving remainder 5 and 8 respectively.

Solution:

It is given that on dividing 70 by the required number, there is a remainder 5.

This means that 70 – 5 = 65 is exactly divisible by the required number.

Similarly, 125 – 8 = 117 is also exactly divisible by the required number.

65 = 5 × 13

117 = 32 × 13

HCF = 13

Required number = 13

Q3. Find the largest number that will divide 398, 436 and 542 leaving remainders 7, 11, and 15 respectively.

Solution:

Algorithm

398 – 7 = 391, 436 – 11 = 425, 542 – 15 = 527

HCF of 391, 425, 527 = 17

Q4. A number N when divided by 14 gives the remainder 5. What is the remainder when the same number is divided by 7?

Solution:

5, because 14 is multiple of 7.

Therefore, remainder in both cases are same.

Q5. What is the HCF of the smallest composite number and the smallest prime number?

Solution:

Smallest composite number = 4

Smallest prime number = 2

So, HCF (4, 2) = 2

Q6. Using Euclid’s Algorithm, find the HCF of 2048 and 960.

Solution:

2048 > 960

Using Euclid’s division algorithm,

2048 = 960 × 2 + 128

960 = 128 × 7 + 64

128 = 64 × 2 + 0

Therefore, the HCF of 2048 and 960 is 64.

Q7. Show that 3√7 is an irrational number.

Solution:

Let us assume, to the contrary, that 3√7 is rational.

That is, we can find coprime a and b (b ≠ 0) such that 3√7 = a/b

Rearranging, we get √7 = a/3b

Since 3, a and b are integers, a/3b is rational, and so √7 is rational.

But this contradicts the fact that √7 is irrational.

So, we conclude that 3√7 is irrational.

Q8. Explain why (17 × 5 × 11 × 3 × 2 + 2 × 11) is a composite number?

Solution:

17 × 5 × 11 × 3 × 2 + 2 × 11 …(i)

= 2 × 11 × (17 × 5 × 3 + 1)

= 2 × 11 × (255 + 1)

= 2 × 11 × 256

Number (i) is divisible by 2, 11 and 256, it has more than 2 prime factors.

Therefore (17 × 5 × 11 × 3 × 2 + 2 × 11) is a composite number.

Q9. Three bells toll at intervals of 9, 12, 15 minutes respectively. If they start tolling together, after what time will they next toll together?

Solution:

9 = 32, 12 = 22 × 3, 15 = 3 × 5

LCM = 22 × 32 × 5 = 4 × 9 × 5 = 180 minutes or 3 hours

They will next toll together after 3 hours.

Q10. Find the HCF and LCM of 306 and 657 and verify that LCM × HCF = Product of the two numbers.

Solution:

306 = 2 × 32 × 17

657 = 32 × 73

HCF = 32 = 9

LCM = 2 × 32 × 17 × 73 = 22338

L.H.S. = LCM × HCF = 22338 × 9 = 201042

R.H.S. = Product of two numbers = 306 × 657 = 201042

L.H.S. = R.H.S.