NCERT Class-10 Maths Chapter-1: Real Numbers

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Exercise-1.1

1. Use Euclid’s division algorithm to find the HCF of :

(i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255

2. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.

3. An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

4. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

[Hint : Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square

each of these and show that they can be rewritten in the form 3m or 3m + 1.]

5. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

Answers:-

2) Let a be any positive integer and b = 6. Then, by Euclid's algorithm, a = 6q+r, for some integer q ≥ 0,

and r = 0, 1, 2, 3, 4, 5, because 0≤r<6.

Now substituting the value of r, we get,

If r = 0, then a 6q

Similarly, for r= 1, 2, 3, 4 and 5, the value of a is 6q+1, 6q+2, 6q+3, 6q+4 and 6q+5, respectively.

If a 6q, 6q+2, 6q+4, then a is an even number and divisible by 2. A positive integer can be either even or odd Therefore, any positive odd integer is of the form of 6q+1, 6q+3 and 6q+5, where q is some integer.

4) Let x be any positive integer and y = 3.

By Euclid's division algorithm, then,

x = 3q + r for some integer q20 and r = 0, 1, 2, as r20 and r < 3.

Therefore, x= 3q, 3q+1 and 3q+2

Now as per the question given, by squaring both the sides, we get,

x² = (3q)2 = 9q² = 3 x 3q²

Let 3q² = m

Therefore, x2= 3m ...................(1)

x²= (3q + 1)² = (3q)2+1²+2x3qx1 = 9q²+1 +6q=3(3q²+2q) +1

Substitute, 3q2+2q = m, to get,

x²= 3m + 1 ...............................(2)

x²= (3q + 2)² = (3q)2+2²+2x3qx2 = 9q² + 4 + 12q = 3 (3q² +4q+1)+1

Again, substitute, 3q2+4q+1= m, to get,

x²= 3m + 1 .............................. (3)

Hence, from equation 1, 2 and 3, we can say that, the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

5)Let x be any positive integer and y= 3.

By Euclid's division algorithm, then,

x = 3q+r, where q20 and r = 0, 1, 2, as r20 and r < 3.

Therefore, putting the value of r, we get,

x = 3q

x = 3q + 1

x = 3q + 2

Now, by taking the cube of all the three above expressions, we get,

Case (i): When r = 0, then,

x²= (3q)³ = 27q³= 9(3q³)= 9m; where m = 3q³

Case (ii): When r = 1, then,

x³ = (3q+1)³= (3q)³ +1³+3x3qx1 (3q+1) = 27q³+1+27q²+9q

Taking 9 as common factor, we get,

x³= 9(3q³+3q2+q)+1

Putting (3q3+3q2 + q) = m, we get,

Putting (3q³+3q2q) = m, we get,

x³ = 9m+1

Case (iii): When r = 2, then,

x³ = (3q+2)³= (3q)³+2³+3x3qx2(3q+2) = 27q³+54q²+36q+8

Taking 9 as common factor, we get,

x³=9(3q³+6q²+4q)+8

Putting (3q³+6q²+4q) = m, we get,

x³ = 9m+8

Therefore, from all the three cases explained above, it is proved that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

Exercise-1.2

1. Express each number as a product of its prime factors:

(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

2. Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.

(i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54

3. Find the LCM and HCF of the following integers by applying the prime factorisation method.

(i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25

4. Given that HCF (306, 657) = 9, find LCM (306, 657).

5. Check whether 6n can end with the digit 0 for any natural number n.

6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

Answers:-

(iv) 5005

By Taking the LCM of 5005, we will get the product of its prime factor. Hence, 5005 5x7x 11 x 13 x1=5x7x 11 x 13

(v) 7429

By Taking the LCM of 7429, we will get the product of its prime factor. Hence, 7429 17 x 19 x 23 x 1 = 17 x 19 x 23

4) As we know that,

HCF X LCM = Product of the two given numbers

Therefore,

9 x LCM = 306 x 657

LCM (306x657)/9 = 22338

Hence, LCM(306,657) = 22338

5) If the number 6" ends with the digit zero (0), then it should be divisible by 5, as we know any number with unit place as 0 or 5 is divisible by 5.

Prime factorization of 6" = (2x3)"

Therefore, the prime factorization of 6" doesn't contain prime number 5.

Hence, it is clear that for any natural number n, 6" is not divisible by 5 and thus it proves that 6" cannot end with the digit 0 for any natural number n.

6) By the definition of composite number, we know, if a number is composite, then it means it has factors other than 1 and itself. Therefore, for the given expression; 7 x 11 x 13 + 13

Taking 13 as common factor, we get, =13(7x11x1+1)= 13(77+1)= 13x78 = 13x3x2x13

Hence, 7 x 11 x 13 + 13 is a composite number.

Now let's take the other number, 7 x 6 x 5 x 4 x 3 x 2 x 1 +5

Taking 5 as a common factor, we get, =5(7x6x4×3×2×1+1)= 5(1008+1) = 5x1009

Hence, 7 x 6 x 5 x 4 x 3 x 2 x 1 +5 is a composite number.

7) Since, Both Sonia and Ravi move in the same direction and at the same time, the method to find the time when they will be meeting again at the starting point is LCM of 18 and 12.

Therefore, LCM(18,12)=2x3x3x2x1=36

Hence, Sonia and Ravi will meet again at the starting point after 36 minutes.

Exercise-1.3

1. Prove that √5 is irrational.

2. Prove that 3 + 2√5 is irrational.

3. Prove that the following are irrationals :

(i)1/√2 (ii) 7√5 (iii) 6 + √2

Answers:-

Exercise-1.4

1. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

(i)13/3125 (ii)17/8 (iii)64/455 (iv)15/1600 (v)29/343 (vi)23/2^3 x 5^2 (vii)129/2^2 x 5^7 x 7^5

(viii)6/15 (ix)35/50 (x)77/210

2. Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

3. The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form p/q, what can you say about the prime factors of q?

(i) 43.123456789 (ii) 0.120120012000120000. . . (iii) 43.123456789123456789123456789.................

Answers:-