NCERT Solutions for Class 10 Maths Chapter 1- Real Numbers

NCERT Solutions Class 10 Maths Chapter 1 Real Numbers are provided here with completed solutions. They solve are provide the NCERT Solutions for Class X 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 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 equations 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 (3q³ + 3q² + q) = 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.

Exercise 1.2

1. Express each number as a product of its prime factors:
   (i) 140
   (ii) 156
   (iii) 3825
   (iv) 5005
   (v) 7429
   Solutions:
   (i) 140
   By Taking the LCM of 140, we will get the product of its prime factor.
   Therefore, 140 = 2 × 2 × 5 × 7 × 1 = 2
   2×5×7
   (ii) 156
   By Taking the LCM of 156, we will get the product of its prime factor.
   Hence, 156 = 2 × 2 × 13 × 3 × 1 = 2
   2× 13 × 3
   (iii) 3825
   By Taking the LCM of 3825, we will get the product of its prime factor.
   Hence, 3825 = 3 × 3 × 5 × 5 × 17 × 1 = 3
   2×52×17
   (iv) 5005
   By Taking the LCM of 5005, we will get the product of its prime factor.
   Hence, 5005 = 5 × 7 × 11 × 13 × 1 = 5 × 7 × 11 × 13
   (v) 7429
   By Taking the LCM of 7429, we will get the product of its prime factor.
   Hence, 7429 = 17 × 19 × 23 × 1 = 17 × 19 × 23

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
   Solutions:
   (i) 26 and 91
   Expressing 26 and 91 as product of its prime factors, we get,
   26 = 2 × 13 × 1
   91 = 7 × 13 × 1
   Therefore, LCM (26, 91) = 2 × 7 × 13 × 1 = 182
   And HCF (26, 91) = 13
   Verification
   Now, product of 26 and 91 = 26 × 91 = 2366
   And Product of LCM and HCF = 182 × 13 = 2366
   Hence, LCM × HCF = product of the 26 and 91.
   (ii) 510 and 92
   Expressing 510 and 92 as product of its prime factors, we get,
   510 = 2 × 3 × 17 × 5 × 1
   92 = 2 × 2 × 23 × 1
   Therefore, LCM(510, 92) = 2 × 2 × 3 × 5 × 17 × 23 = 23460
   And HCF (510, 92) = 2
   Verification
   Now, product of 510 and 92 = 510 × 92 = 46920
   And Product of LCM and HCF = 23460 × 2 = 46920
   Hence, LCM × HCF = product of the 510 and 92.
   (iii) 336 and 54
   Expressing 336 and 54 as product of its prime factors, we get,
   336 = 2 × 2 × 2 × 2 × 7 × 3 × 1
   54 = 2 × 3 × 3 × 3 × 1
   Therefore, LCM(336, 54) = 2
   4 × 3
   3 × 7 = 3024
   And HCF(336, 54) = 2×3 = 6
   Verification
   Now, product of 336 and 54 = 336 × 54 = 18,144
   And Product of LCM and HCF = 3024 × 6 = 18,144
   Hence, LCM × HCF = product of the 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
   Solutions:
   (i) 12, 15 and 21
   Writing the product of prime factors for all the three numbers, we get,
   12=2×2×3
   15=5×3
   21=7×3
   Therefore,
   HCF(12,15,21) = 3
   LCM(12,15,21) = 2 × 2 × 3 × 5 × 7 = 420
   (ii) 17, 23 and 29
   Writing the product of prime factors for all the three numbers, we get,
   17=17×1
   23=23×1
   29=29×1
   Therefore,
   HCF(17,23,29) = 1
   LCM(17,23,29) = 17 × 23 × 29 = 11339
   (iii) 8, 9 and 25
   Writing the product of prime factors for all the three numbers, we get,
   8=2×2×2×1
   9=3×3×1
   25=5×5×1
   Therefore,
   HCF(8,9,25)=1
   LCM(8,9,25) = 2×2×2×3×3×5×5 = 1800
4. Given that HCF (306, 657) = 9, find LCM (306, 657).
   Solutions: As we know that,
   HCF×LCM=Product of the two given numbers
   Therefore,
   9 × LCM = 306 × 657
   LCM = (306×657)/9 = 22338
   Hence, LCM(306,657) = 22338

5. Check whether 6ⁿ can end with the digit 0 for any natural number n.
   Solutions: If the number 6ⁿ 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 6ⁿ = (2×3)n
   Therefore, the prime factorization of 6ⁿ doesn’t contain prime number 5.
   Hence, it is clear that for any natural number n, 6n is not divisible by 5 and thus it proves that 6ⁿ cannot 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.
   Solutions: 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 × 11 × 13 + 13
   Taking 13 as common factor, we get,
   =13(7×11×1+1) = 13(77+1) = 13×78 = 13×3×2×13
   Hence, 7 × 11 × 13 + 13 is a composite number.
   Now let’s take the other number,
   7 × 6 × 5 × 4 × 3 × 2 × 1 + 5
   Taking 5 as a common factor, we get,
   =5(7×6×4×3×2×1+1) = 5(1008+1) = 5×1009
   Hence, 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 is a composite number.
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?
   Solutions: 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) = 2×3×3×2×1=36
   Hence, Sonia and Ravi will meet again at the starting point after 36 minutes.

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