**Q5 ** Use Euclid’s division lemma to show that the cube of any positive integer is of the form

9m, 9m + 1 or 9m + 8.

Let x be any positive integer.
It can be written in the form 3q + r where and r = 0, 1 or 2
Case 1:
For r = 0 we have
x3 = (3q)3
x3 = 27q3
x3 = 9(3q3)
x3 = 9m
Case 2:
For r = 1 we have
x3 = (3q+1)3
x3 = 27q3 + 27q2 + 9q + 1
x3 = 9(3q3 + 3q2 +q) + 1
x3 = 3m + 1
Case 3:
For r = 2 we have
x3 = (3q+2)3
x3 = 27q3 + 54q2 + 36q + 8
x3 = 9(3q3 + 6q2 +4q) + 8
x3 = 3m + 8
Hence proved.

**Q4 ** 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 x 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.]

Let x be any positive integer.
It can be written in the form 3q + r where and r = 0, 1 or 2
Case 1:
For r = 0 we have
x2 = (3q)2
x2 = 9q2
x2 = 3(3q2)
x2 = 3m
Case 2:
For r = 1 we have
x2 = (3q+1)2
x2 = 9q2 + 6q +1
x2 = 3(3q2 + 2q) + 1
x2 = 3m + 1
Case 3:
For r = 2 we have
x2 = (3q+2)2
x2 = 9q2 + 12q +4
x2 = 3(3q2 + 4q + 1) + 1
x2 = 3m + 1
Hence proved.

**Q3** 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?

The maximum number of columns in which they can march = HCF (32, 616)
Since 616 > 32, applying Euclid's Division Algorithm we have
Since remainder 0 we again apply Euclid's Division Algorithm
Since 32 > 8
Since remainder = 0 we conclude, 8 is the HCF of 616 and 32.
The maximum number of columns in which they can march is 8.

**Q2 **Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is

some integer.

Let p be any positive integer. It can be expressed as
p = 6q + r
where and
but for r = 0, 2 or 4 p will be an even number therefore all odd positive integers can be written in the form 6q + 1, 6q + 3 or 6q + 5.

**Q1 (3) ** Use Euclid’s division algorithm to find the HCF of : 867 and 255

867 > 225. Applying Euclid's Division algorithm we get
since remainder 0 we apply the algorithm again.
since 255 > 102
since remainder 0 we apply the algorithm again.
since 102 > 51
since remainder = 0 we conclude the HCF of 867 and 255 is 51.

**Q1 (2)** Use Euclid’s division algorithm to find the HCF of : 196 and 38220

38220 > 196. Applying Euclid's Division algorithm we get
since remainder = 0 we conclude the HCF of 38220 and 196 is 196.

**Q1 (1) ** Use Euclid’s division algorithm to find the HCF of : 135 and 225

225 > 135. Applying Euclid's Division algorithm we get
since remainder 0 we again apply the algorithm
since remainder 0 we again apply the algorithm
since remainder = 0 we conclude the HCF of 135 and 225 is 45.

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