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

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

1. (i) Here, 225 > 135
Applying Euclid’s division lemma to 225 and 135, we get
225 = (135 × 1) + 90
Since, 90 ≠ 0, therefore, applying Euclid’s division lemma
to 135 and 90, we get
135 = (90 × 1) + 45
Since, 45 ≠ 0
Applying Euclid’s division lemma to 90 and 45,
we get 90 = (45 × 2) + 0
Here, remainder, r = 0, when divisor is 45.
HCF of 225 and 135 is 45.
(ii) Here, 38220 > 196
Applying Euclid’s division lemma, we get
38220 = (196 × 195) + 0
Here, r = 0, when divisor is 196.
HCF of 38220 and 196 is 196.
(iii) Here, 867 > 255
Applying Euclid’s division lemma, we get
867 = (255 × 3) + 102,
255 = (102 × 2) + 51,
102 = (51× 2) + 0
Here remainder = 0, when divisor is 51.
HCF of 867 and 255 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

2. Let us consider a positive odd integer as ‘a’.
On dividing ‘a’ by 6, let ‘q’ be the quotient and ‘r’ be the
remainder.
\ Using Euclid’s division lemma, we get
a = 6q + r, where 0 ≤ r < 6
a = 6q + 0 = 6q or a = 6q + 1
or a = 6q + 2 or a = 6q + 3
or a = 6q + 4 or a = 6q + 5
But, a = 6q, a = 6q + 2, a = 6q + 4 are even values of ‘a’.
[Q 6q = 2(3q) = 2m1, 6q + 2 = 2(3q + 1) = 2m2,
6q + 4 = 2(3q + 2) = 2m3]
Being an odd integer, we have
a = 6q + 1 or a = 6q + 3 or a = 6q + 5

### 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?

3. 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 division lemma to 616 and 32, we get
616 = (32 × 19) + 8,
32 = (8 × 4) + 0
Here, remainder = 0, when divisor is 8.
\ HCF of 616 and 32 is 8.
Hence, the required number of maximum columns = 8.

### 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

4. Let us consider an arbitrary positive integer as
‘x’. On dividing x by 3 and applying Euclid’s division
lemma, we get that x is of the form,
3q, (3q + 1) or (3q + 2)
For x = 3q, we have
x2= (3q)2

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

5. Let us consider an arbitrary positive integer x such that it is of the form 3q, (3q + 1), or (3q + 2).
For x = 3q

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