**Answer
1** :

**i. 135 and 225**

**ii. 196 and 38220**

**iii. 867 and 225**

**Solutions: **

** Q i. 135 and 225**

i. 135 and 225

As you can see, from the question 225 isgreater than 135. Therefore, by Euclid’s division algorithm, we have,

225 = 135 × 1 + 90

Now, remainder 90 ≠ 0, thus again usingdivision lemma for 90, we get,

135 = 90 × 1 + 45

Again, 45 ≠ 0, repeating the above step for45, we get,

90 = 45 × 2 + 0

The remainder is now zero, so our method stopshere. 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.

**Q ii. 196 and 38220**

ii. 196 and 38220

In this given question, 38220>196,therefore the by applying Euclid’s division algorithm and taking 38220 asdivisor, 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.

**Q iii. 867 and 225**

iii. 867 and 225

As we know, 867 is greater than 225. Let usapply now Euclid’s division algorithm on 867, to get,

867 = 225 × 3 + 102

Remainder 102 ≠ 0, therefore taking 225 asdivisor and applying the division lemma method, we get,

225 = 102 × 2 + 51

Again, 51 ≠ 0. Now 102 is the new divisor, sorepeating the same step we get,

102 = 51 × 2 + 0

The remainder is now zero, so our procedurestops 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.

**Show that any positive odd integer is of theform 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.**

**Answer
2** :

**An army contingent of 616 members is to marchbehind an army band of 32 members in a parade. The two groups are to march inthe same number of columns. What is the maximum number of columns in which theycan march?**

**Answer
3** :

**Solution:**

Given,

Number of army contingent members=616

Number of army band members = 32

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 newdivisor, 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 whichthey can march is 8.

**Use Euclid’s division lemma to show that thesquare of any positive integer is either of the form 3m or 3m + 1 for someinteger m.**

**Answer
4** :

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

**Answer
5** :

**Solution:**

Let x be any positive integer and y = 3.

By Euclid’s division algorithm

Now

x = 3q+r, where q≥0 and r = 0, 1, 2, as r ≥ 0and r < 3.

Then, 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 aboveexpressions.

**Case (i):** Whenr = 0

Then,

x^{2}= (3q)^{3} = 27q^{3}=9(3q^{3})= 9m; where m = 3q^{3}

**Case (ii):** Whenr = 1

Then,

x^{3} = (3q+1)^{3} =(3q)^{3 }+1^{3}+3×3q×1(3q+1) = 27q^{3}+1+27q^{2}+9q

Taking 9 as common above factor

We get,

x^{3 }= 9(3q^{3}+3q^{2}+q)+1

Putting = m

We get,

Putting (3q^{3}+3q^{2+}q) = m,we get ,

x^{3} = 9m+1

**Case (iii): **When r = 2

Then,

x^{3} = (3q+2)^{3}

=(3q)^{3}+2^{3}+3×3q×2(3q+2)

=27q^{3}+54q^{2}+36q+8

Taking 9 as common above factor

We get,

x^{3}=9(3q^{3}+6q^{2}+4q)+8

Putting (3q^{3}+6q^{2}+4q) = m

We get ,

x^{3} = 9m+8

Therefore, from all the three cases explainedabove, it is proved that the cube of any positive integer is of the form 9m, 9m+ 1 or 9m + 8.

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