NCERT Class 10 Maths Solutions Chapter - 1 Real Numbers

Ex - 1.1

Q.1  Use Euclid's division algorithm to find the HCF of:
(i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255

Solution

(i) 135 and 225 

Step 1:  Since 225 > 135, apply Euclid's division lemma, to a =225 and b=135 to find q and r   such that 225 = 135q+r, 0  r<135 

On dividing 225 by 135 we get quotient as 1 and remainder as 90 
i.e 225 = 135 x 1 + 90 

Step 2: Remainder r which is 90  0, we apply Euclid's division lemma to b =135 and r = 90 to find whole numbers q and r such that 
135 = 90 x q + r,  0  r<90
On dividing 135 by 90 we get quotient as 1 and remainder as 45
i.e 135 = 90 x 1 + 45

Step 3:  Again remainder r = 45  0 so we apply Euclid's division lemma to b =90 and r = 45 to find q and r   such that 
90 = 90 x q + r,    0  r<45
On dividing 90 by 45 we get quotient as 2 and remainder as 0 
i.e 90 = 2 x 45 + 0

Step 4: Since the remainder is zero, the divisor at this stage will be HCF of (135, 225).  

Since the divisor at this stage is 45, therefore, the HCF of 135 and 225 is 45. 

(ii)    196 and 38220
Step 1:  Since 38220 > 196, apply Euclid's division lemma
to a =38220 and b=196  to find  whole numbers q and r   such that
 38220 = 196 q + r, 0  r < 196
On dividing 38220 we get quotient as 195 and remainder r as 0 
i.e 38220 = 196 x 195 + 0
Since the remainder is zero, divisor at this stage will be HCF 
Since divisor at this stage is 196 , therefore, HCF of 196 and 38220 is 196.

NOTE: HCF( a,b) = a if a is a factor of  b. Here, 196 is a factor of  38220 so HCF is 196. 
(iii)    867 and 255 

Step 1: Since 867 > 255, apply Euclid's division lemma, to a =867 and b=255 to find q and r   such that 867 = 255q + r, 0  r<255
On dividing 867 by 255 we get quotient as 3 and remainder as 102
i.e 867 = 255 x 3 + 102 

Step 2: Since remainder 102  0, we apply the division lemma to a=255 and b= 102 to find whole numbers q and r such that 
255 = 102q + r where 0  r<102
On dividing 255 by 102 we get quotient as 2 and remainder as 51 
i.e 255 = 102 x 2 + 51

Step 3: Again remainder 51 is non zero, so we apply the division lemma to a=102 and b= 51  to find whole numbers q and r such that 
102 = 51 q + r where 0  r < 51 

On dividing 102 by 51 quotient is 2 and remainder is 0
i.e 102 = 51 x 2 + 0
Since the remainder is zero, the divisor at this stage is the HCF
Since the divisor at this stage is 51,therefore, HCF of 867 and 255 is 51. 

Concept Insight: To crack such problem remember to apply the Euclid's division Lemma which states that "Given positive integers a and b, there exists unique integers q and r satisfying a = bq + r, where 0  r < b" in the correct order. 

Here, a > b. 

Euclid's algorithm works since Dividing 'a' by 'b', replacing 'b' by 'r' and 'a' by 'b' and repeating the process of division till remainder 0 is reached, gives  a number which divides a and b exactly. 

i.e    HCF(a,b) =HCF(b,r)

Note that do not find the HCF using prime factorisation in this question when the method is specified and do not skip steps.

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