1. Use Euclid’s division algorithm to find the HCF of :
(i) 135 and 225
Solution:
Step 1.) Start with the larger integer, that is, 225. Then we use Euclid’s lemma to get:
225= 135x1 + 90

Step 2.) Now consider the divisor 135 and the remainder 90, and apply the division lemma to get
135=90x1+45

Step 3.)Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

90=45x2+0

Answer: HCF of (135,225) is 45.

(ii) 196 and 38220
Solution:
Step 1.) Start with the larger integer, that is,38220. Then we use Euclid’s lemma to get:
38220 = 196x195+0
Step 2.)Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

Since the reminder is 0,divisor at this stage will be the required HCF

Answer: HCF of (196,38220) is 196


(iii) 867 and 255
Solution
Step 1.) Start with the larger integer, that is,867. Then we use Euclid’s lemma to get:
867= 255x3 + 102

Step 2.) Now consider the divisor 255 and the remainder 102, and apply the division lemma to get
255=102x2 + 51 

Step 3.)Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

102= 51x2 +0
Answer: HCF of (867,255) is 51

Question 2.


Question 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:
Since, an army contingent of 616 members has to march behind an army band of 32 member in a parade, we have
to find the highest commom factor which can divide 616 and 32 both . So, we have to find the HCF of 616 and 32 .
So use Euclid's lemma method to find HCF.

616= 32 x 19 + 8

Repeat the step till the remainder is zero.The divisor at this stage will be the required solution.

32=8x4 + 0
HCF of (616,32) is 8 

Answer:So maximum number of columns in which they can march is 8.

Additional requirement. If the above question, requires you to find the number of rows,
in which they can be arranged, we first have to find the number of columns in which they can march as done
earlier and then find the number of rows in which they can be arranged.
An army band of 32 members can be arranged in ${32/8 = 4}$ numbers of rows.
and An army contingent of 616 members can be arranged in ${616/8= 77 }$ numbers of rows.
Therefore total numbers of rows of the parade contingent = 4+77=81 numbers of rows.