Three sets of Physics, Chemistry and Biology books have to be stacked in such a way that all the books are stored topic-wise and the height of each stack is the same. The number of Physics books is 105, the number of Chemistry books is 140 and the number of Biology books is 175. Assuming that the books are of the same thickness, determine the number of stacks of Physics, Chemistry and Biology books.

Solution

In order to arrange the books as required, we have to find the largest number that divides 105, 140 and 175 exactly.

Clearly, required number is the HCF of 105, 140 and 175.

Case I :

I. Finding HCF of 105 and 140 by applying Euclid’s division lemma, we get

II. Since, the remainder 35 ≠ 0, we apply division lemma to get

Since, the remainder at this stage is zero, so the divisor i.e., 35 at this stage is the HCF of 105 and 140.

Case II :

I. Finding the HCF of 35 and 175 by applying Euclid’s division lemma, we get

II. Since the remainder at this stage is zero, so the divisor i.e., 35 at this stage is the HCF of 35 and 175.

Thus, HCF of 105, 140) and 175 is 35.

Now,

Number of stacks of Physics books
$equals fraction numerator Number space of space Physics space books over denominator NMumber space of space books space in space each space stack end fraction equals space 105 over 35 equals 3$
Number of stacks of Chemistry books
$equals fraction numerator Number space of space Chemistry space books over denominator Number space of space books space ineach space stack end fraction$
$equals 140 over 35 equals 4$
Number of stacks of Biology books
$equals fraction numerator Number space of space Biology space books over denominator Number space of space books space in space each space stack end fraction$

$equals 175 over 35 equals 5.$

Some More Questions From Real Numbers Chapter

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Real Numbers

Some More Questions From Real Numbers Chapter

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.

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

Express each number as a product of its prime factors: (i) 140

Express each number as a product of its prime factors: (ii) 156

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

Express each number as a product of its prime factors: (iv) 5005

Express each number as a product of its prime factors: (v) 7429

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (i) 26 and 91

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (ii) 510 and 92

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (iii) 336 and 54

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