Exercise 1.1

 

SET I

 

1.             Use Euclid’s algorithm to find the HCF of 4052 and 12576.

2.             Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer.

3.             Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.

4.             A sweetseller has 420 kaju barfis and 130 badam barfis. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the number of that can be placed in each stack for this purpose?

 

SET II

 

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 column 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.

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

 

SET III

 

1.             Prove that the product of two consecutive positive integers is divisible by 2.

2.             If a and b are two odd positive integers such that a > b, the prove that one of the two numbers and  is odd and other is even.

3.             Show that the square of an odd positive integer is of the form 8q + 1, for some integer q.

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

5.             Show that n² – 1 is divisible by 8, if n is an odd positive integer.

6.             Prove that if x and y are both odd positive integers, then x² + y² is even but not divisible by 4.

7.             Show that the square of an odd positive integer is of the form 8m + 1, for some whole number m.

8.             Prove that n² – n is divisible by 2 for every positive integer n.

9.             Prove that one of every three consecutive positive integers is divisible by 3.

10.          Show that any positive odd integer is of the form 8q+1, or 8q+3, or 8q+5 or 8q+7, where q is some integer.

11.          Write whether every positive integer can be of the form 4q + 2, where q is an integer. Justify your answer.

12.          “The product of two consecutive positive integers is divisible by 2”. Is this statement true or false? Give reasons.

13.          “The product of three consecutive positive integers is divisible by 6”. Is this statement true or false”? Justify your answer.

14.          Write whether the square of any positive integer can be of the form 3m + 2, where m is a natural number. Justify your answer.

15.          A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify your answer.

16.          Show that the square of any positive integer is either of the form 4q or 4q + 1 for some integer q.

17.          Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m.

18.          Show that the square of any positive integer cannot be of the form 5q + 2 or 5q+3 for any integer q.

19.          Show that the square of any positive integer cannot be of the form 6m + 2 or 6m+5 for any integer m.

20.          Show that the square of any odd integer is of the form 4q + 1, for some integer q.

21.          If n is an odd integer, then show that n² – 1 is divisible by 8.

22.          Show that the square of any positive integer is of the form 3m or 3m + 1 for some integer m.

23.          Show that one and only one out of n, n + 2, or n + 4 is divisible by 3, where n is any positive integer.

24.          Prove that the square of any positive integer is either of the form 3m or 3m+1 but not of the form 3m + 2.

25.          Show that the square of any odd integer is of the form 5q, 5q + 1, 5q +4 for some integer q.

26.          Prove that if a positive integer is of the form 6q + 5, then it is of the form 3q + 2 for some integer q, but not conversely.

27.          Prove that the square of any positive integer of the form 5q + 1 is of the same form.

28.          Prove that the product of three consecutive positive integers is divisible by 6.

29.          For any positive integer n, prove that n³ – n divisible by 6.

 

Exercise 1.2

 

SET I

 

1.             Consider the numbers 4ⁿ, where n is a natural number. Check whether there is any value of n for which 4ⁿ ends with the digit zero.

2.             Find the LCM and HCF of 6 and 20 by the prime factorisation method.

3.             Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.

4.             Find the HCF and LCM of 6, 72 and 120, using the prime factorization method.

SET II

 

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 factorization 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).

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?

 

SET III

 

1.             Renu purchases two bags of fertiliser of weights 75 kg and 69 kg. Find the maximum value of weight which can measure the weight of the fertiliser exact number of times.

2              Three boys step off together from the same spot. Their steps measure 63 cm, 70 cm and 77 cm respectively. What is the minimum distance each should cover so that all can cover the distance in complete steps?

3.             The length, breadth and height of a room are 825 cm, 675 cm and 450 cm respectively. Find the longest tape which can measure the three dimensions of the room exactly.

4.             Determine the smallest 3-digit number which is exactly divisible by 6, 8 and 12.

5.             Determine the greatest 3-digit number exactly divisible by 8, 10 and 12.

6.             The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 7 a.m., at what time will they change simultaneously again?

7.             Three tankers contain 403 litres, 434 litres and 465 litres of diesel respectively. Find the maximum capacity of a container that can measure the diesel of the three containers exact number of times.

8.             Find the least number which when divided by 6, 15 and 18 leave remainder 5 in each case.

9.             Find the smallest 4-digit number which is divisible by 18, 24 and 32.

10.          Can the number 6ⁿ, n being a natural number, end with the digit 5? Give reasons.

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

12.          Find the [HCF × LCM] for the numbers 100 and 190.

 

 

 

 

Exercise 1.3

 

SET I

 

1.             Prove that √2 is irrational.

2.             Prove that √3 is irrational.

3.             Show that 5 – √3 is irrational.

4.             Show that 3√2 is irrational.

 

SET II

 

1.             Prove that √5 is irrational.

2.             Prove that 3 + 2√5 is irrational.

3.             Prove that the following are irrationals:

(i)                    (ii)           7√5         (iii)          6 + √2

 

SET III

 

1.             Define irrational numbers.

2.             Prove that √2 + √3 is irrational.

 

Exercise 1.4

 

SET II

 

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)                              

(ii)                  

(iii)               

(iv)             

(v)                

(vi)             

(vii)                        

(viii)                  

(ix)                  

(x)              

 

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 , what can you say about the prime factors of q?

(i)            43.123456789     

(ii)          0.120120012000120000...

(iii)        

NCERT Class X Mathematics Chapter 1: Real Numbers

 

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

i. 135 and 225

ii. 196 and 38220

iii. 867 and 225

Solutions:

i. 135 and 225

As you can see, from the question 225 is greater than 135. Therefore, by Euclid’s division algorithm, we have,

225 = 135 × 1 + 90

Now, remainder 90 ≠ 0, thus again using division lemma for 90, we get,

135 = 90 × 1 + 45

Again, 45 ≠ 0, repeating the above step for 45, we get,

90 = 45 × 2 + 0

The remainder is now zero, so our method stops here. Since, in the last step, the divisor is 45, therefore, HCF (225,135) = HCF (135, 90) = HCF (90, 45) = 45.

Hence, the HCF of 225 and 135 is 45.

ii. 196 and 38220

In this given question, 38220>196, therefore the by applying Euclid’s division algorithm and taking 38220 as divisor, we get,

38220 = 196 × 195 + 0

We have already got the remainder as 0 here. 

Therefore, HCF(196, 38220) = 196.

Hence, the HCF of 196 and 38220 is 196.

iii. 867 and 225

As we know, 867 is greater than 225. Let us apply now Euclid’s division algorithm on 867, to get,

867 = 225 × 3 + 102

Remainder 102 ≠ 0, therefore taking 225 as divisor and applying the division lemma method, we get,

225 = 102 × 2 + 51

Again, 51 ≠ 0. Now 102 is the new divisor, so repeating the same step we get,

102 = 51 × 2 + 0

The remainder is now zero, so our procedure stops here. Since, in the last step, the divisor is 51, therefore, HCF (867,225) = HCF(225,102) = HCF(102,51) = 51.

Hence, the HCF of 867 and 225 is 51.

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

Solution:

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.

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?

Solution:

Given,

Number of army contingent members=616

Number of army band members = 32

If the two groups have to march in the same column, we have to find out the highest common factor between the two groups. HCF(616, 32), gives the maximum number of columns in which they can march.

By Using Euclid’s algorithm to find their HCF, we get,

Since, 616>32, therefore,

616 = 32 × 19 + 8

Since, 8 ≠ 0, therefore, taking 32 as new divisor, we have,

32 = 8 × 4 + 0

Now we have got remainder as 0, therefore, HCF (616, 32) = 8.

Hence, the maximum number of columns in which they can march is 8.

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.

Solutions:

Let x be any positive integer and y = 3.

By Euclid’s division algorithm, then,

x = 3q + r for some integer q≥0 and r = 0, 1, 2, as r ≥ 0 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 × 3q²

Let 3q² = m

Therefore, x²= 3m ……………………..(1)

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

Substitute, 3q²+2q = m, to get,

x²= 3m + 1 ……………………………. (2)

x²= (3q + 2)² = (3q)²+2²+2×3q×2 = 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. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

Solution:

Let x be any positive integer and y = 3.

By Euclid’s division algorithm, then,

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

Therefore, putting the value of r, we get,

x = 3q

or

x = 3q + 1

or

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+3×3q×1(3q+1) = 27q³+1+27q²+9q

Taking 9 as common factor, we get,

x³ = 9(3q³+3q²+q)+1

Putting = m, we get,

Putting (3q³+3q²⁺q) = m, we get ,

x³ = 9m+1

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

x³ = (3q+2)³= (3q)³+2³+3×3q×2(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.