CBSE Class 10 Mathematics Real Numbers Notes And Examples

EUCLID’S DIVISION ALGORITHM :

If ‘a’ and ‘b’ are positive integers such that a = bq + r, then every common divisor of ‘a’ and ‘b’ is a common divisor of ‘b’ and ‘r’ and vice-versa.

Illustration-4 Use Euclid’s division algorithm to find the H.C.F. of 196 and 38318.

Answer- Applying Euclid’s division lemma to 196 and 38318.

38318 = 195 × 196 + 98

196 = 98 × 2 + 0

The remainder at the second stage is zero. So, the H.C.F. of 38318 and 196 is 98.

Illustration-5 If the H.C.F. of 657 and 963 is expressible in the form 657x + 963 × (-15), find x.

Answer- Applying Euclid’s division lemma on 657 and 963.

963 = 657 × 1 + 306

657 = 306 × 2 + 45

306 = 45 × 6 + 36

45 = 36 × 1 + 9

36 = 9 × 4 + 0

So, the H.C.F. of 657 and 963 is 9.

Given : 657x + 963 × (-15) = H.C.F. of 657 and 963.

657 x + 963 × (-15) = 9

657 x = 9 + 963 × 15

657 x = 14454

x = 14454/657 = 22.
 
Illustration-6 What is the largest number that divides 626, 3127 and 15628 and leaves remainders of 1, 2 and 3 respectively.

Answer- Clearly, the required number is the H.C.F. of the number 626 - 1 = 625, 3127 - 2 3125 and 15628 - 3 = 15625.

15628 - 3 = 15625.

Using Euclid’s division lemma to find the H.C.F. of 625 and 3125.

3125 = 625 × 5 + 0

Clearly, H.C.F. of 625 and 3125 is 625.

Now, H.C.F. of 625 and 15625

15625 = 625 × 25 + 0

So, the H.C.F. of 625 and 15625 is 625.

Hence, H.C.F. of 625, 3125 and 15625 is 625.

Hence, the required number is 625.

Illustration-7 144 cartons of coke cans and 90 cartons of Pepsi cans are to be stacked is a canteen. If each stack is of same height and is to contains cartons of the same drink, what would be the greatest number of cartons each stack would have ?

Answer- In order to arrange the cartons of the same drink is the same stack, we have to find the greatest number that divides 144 and 90 exactly. Using Euclid’s algorithm, to find the H.C.F. of 144 and 90.

144 = 90 × 1 + 54

90 = 54 × 1 + 36

54 = 36 × 1 + 18

36 = 18 × 2 + 0

So, the H.C.F. of 144 and 90 is 18.

Number of cartons in each stack = 18.

FUNDAMENTAL THEOREM OF ARITHMETIC :

Every composite number can be expressed as a product of primes, and this factorisation is unique, except for the order in which the prime factors occurs.

Illustration-11 In a morning walk three persons step off together, their steps measure 80 cm, 85 cm and 90 cm respectively. What is the minimum distance each should walk so that they can cover the distance in complete steps ?

Answer- Required minimum distance each should walk so, that they can cover the distance in complete step is the L.C.M. of 80 cm, 85 cm and 90 cm

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