**2.** Find the **LCM and HCF** of the following pairs of integers and verify that LCM x HCF=product of the two numbers.

**(i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54**

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by Jashan

**2.** Find the **LCM and HCF** of the following pairs of integers and verify that LCM x HCF=product of the two numbers.

**(i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54**

by Jashan

**1.** Express each number as a product of its prime factors.

**(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429**

by Jashan

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

**Solution:**

Let **a** be any positive integer and **b=3**.

According to **Euclid's division lemma**, we can say that

Therefore, possible values of a are:

or

by Jashan

**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 x be any positive integer then it is of the form . Now square each of these and show that they can be rewritten in the form 3m or 3m+1]

**Solution:**

Let x be any positive integer and b=3.

According to **Euclid's division lemma**, we can say that

Therefore, all possible values of **x** are:

or

by Jashan

**3. Any 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?**

**Solutions:**

HCF of 616 and 32 would be equal to maximum number of columns in which they can march.

To find HCF, we can use **Euclid's division algorithm**, we apply **Euclid's division lemma** to given numbers **c and d,** to find whole numbers **q and r** such that