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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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
1. Express each number as a product of its prime factors.
(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429
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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
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
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