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MP Board Class 10th Maths Chapter 1 Real Numbers Ex 1.1 Solutions

In this article, we will share MP Board Class 10th Maths Book Solutions Chapter 1 Real Numbers Ex 1.1 Pdf, These solutions are solved subject experts from the latest edition books. English Solutions here .

MP Board Class 10th Maths Solutions Chapter 1 Real Numbers Ex 1.1

Question 1.
Use Euclid’s division algorithm to find the HCF of
(i) 135 and 225
(ii) 196 and 38220
(iii) 867 and 255

Solution:
(i) HCF of 135 and 225
Applying the Euclid’s lemma to 225 and 135, (where 225 > 135), we get
225 = (135 × 1) + 90, since 90 ≠ 0, therefore, applying the Euclid’s lemma to 135 and
90, we get 135 = (90 × 1) + 45
But 45 ≠ 0
∴ Applying Euclid’s lemma to 90 and 45, we get 90 = (45 × 2) + 0
Here, r = 0, so our procedure stops. Since, the divisor at the last step is 45,
∴ HCF of 225 and 135 is 45.

(ii) HCF of 196 and 38220

We start dividing the larger number 38220 by 196, we get
38220 = (196 × 195) + 0
Here, r = 0
∴ HCF of 38220 and 196 is 196.

(iii) HCF of 867 and 255 Here, 867 > 255
∴ Applying Euclid’s Lemma to 867 and 255, we get
867 = (255 × 3) + 102, 102 ≠ 0
∴ Applying Euclid’s Lemma to 255 and 102, we get
255 = (102 × 2) + 51, 51 ≠ 0
∴ Applying Euclid’s Lemma to 102 and 51, we get
102 = (51 × 2) + 0, r = 0
∴ HCF of 867 and 255 is 51.

Questions 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 numbers of columns. What is the maximum number of columns in which they can march?

Solution: Total number of members 616

∴The total number of members are to march behind an army band of 32 members is HCF of 616 and 32. i.e., HCF of 616 and 32 is equal to the maximum number of columns such that the two groups can march in the same number of columns.

∴Applying Euclid’s lemma to 616 and 32, we get 616 = (32✖19) +8, ≠0

Again, applying Euclid’s lemma to 32 and 8, we get 32 = (8✖4) + 0, r=0

∴Hence, the required number of maximum columns = 8.

Ex 1.1 Question 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 ✖ 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)

Solution:

Let ‘a’ be an Positive integer.

a=bq + r (where b=30

a = 3q + r (where0 ≤ r ∠ 3)

i) if r = 0 a = 3q

a2 = (3q)2 = 9q2

a2 = 3(3q)2 [m = 3q2]

a2 =3m

[m is an Positive integer]

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