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Class 10 NCERT Solutions- Chapter 1 Real Numbers – Exercise 1.1

  • Last Updated : 05 Jun, 2021

Question 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

We need to apply Euclid’s algo, for finding HCF, now First Step to take Divisor ,we always take Divisor as smaller than Dividend. 
Divisor=135 
Dividend=225 
Quotient=225/135=1(quotient is an integer value) 
Euclid’s Div. Algo: 225=135*1+90 
Since 90!=0 that means We have to again apply division lemma for remainder:90,Now Remainder Will become Divisor and Divisor 
will become Dividend. 
Euclid’s Div. Algo: 135=90*1+45 
Again we Need to apply Division lemma because Yet remainder is not equal to 0 (45!=0) and just similar to prev. ways we will get 
Divisor =45 and Dividend=90 
Euclid’s Div. Algo:90=45*2+0 
Here Remainder=0 that means We need to stop here, 
When Remainder = 0 Then HCF = Divisor => HCF(225,135)=HCF(135,90) 
=HCF(90,45) 
=45.

ii)196 and 38220

Divisor=196 
Dividend=38220 
Quotient=38220/196=195 
Euclid’s Div. Algo: 38220=196*195+0 
Remainder=0,We don’t need to apply Division lemma further, 
HCF(38220,196)=196



iii) 867 and 255

Divisor=255 
Dividend=867 
Quotient=867/225=3 
Euclid’s Div. algo: 867=255*3+102 
Remainder = 102 (!=0) That means Again we need to apply Division lemma method, Divisor=102,Dividend=255,Quotient=255/102=2 
Euclid’s Div. algo: 255=102*2+51 
Again Remainder =51(!=0), We need to apply Division lemma method again, Divisor=51,Dividend=102,Quotient=102/51=2 
Euclid’S Div. algo:102=51*2+0 
Remainder =0,We need to stop here , 
HCF(867,255)=HCF(255,102) 
= HCF(102,51) 
= 51

Question 2.Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.

Solution:

**We know that any odd no integer is not divisible by 2, 

When we divide 6q by 2 then it is perfectly divisible by 2 and give result 3q without any remainder,=>6q is even number 

let take any positive integer x and y=6 
as Per Euclid’s Div algo x=6q+r, q>=0 & 0<=r<6 
if r=0 => x=6q, we earlier concluded that 6q is divisible by 2 =>x=6q is positive even integer 

if r=2, 4 =>x=6q+2, x=6q+4 also An positive even integer because 6q is even as well as 2,4 also even and even divided by even gives 
even 

if r=1 
Dividing 6q+1 by 2: 
Dividend=6q+1,Divisor=2 
Quotient = (6q+1)/2 =3q 
as per Euclid’s div algo 6q+1=3q*2+1 
As We Got Remainder = 1 after dividing by 2 that means 6q+1 is odd number, As q>=0 that means 6q+1 is positive odd integer 



if r=3 
Dividing 6q+3 by 2: 
Dividend=6q+3,Divisor=2 
Quotient = (6q+3)/2 =3q+1 
as per Euclid’s div algo 6q+3=(3q+1)*2+1 
As We Got Remainder = 1 after dividing by 2 that means 6q+3 is odd number, As q>=0 that means 6q+3 is positive odd integer 

if r=5 
Dividing 6q+5 by 2: 
Dividend=6q+5,Divisor=2 
Quotient = (6q+5)/2 =3q+2 
as per Euclid’s div algo 6q+5=(3q+2)*2+1 
As We Got Remainder = 1 after dividing by 2 that means 6q+5 is odd number, As q>=0 that means 6q+5 is positive odd integer 
Overall Conclusion : Any positive integer x can be in form 6q+1,6q+3 or 6q+5 if it is odd otherwise it is in form 6q, 6q+2, 6q+4 

Question 3. An 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?

Solution:

Given, 
Total number of army contingent members=616 
Total number of army band members=32 

Given two groups are to march in same number of column then maximum number of columns will be Highest Common Factor between two 
groups ie HCF(616,32) 

Finding HCF(616,32) 
——————— 
Divisor=32,Dividend=616,quotient=616/32=19 
Euclid’s Div. Algo: 616=32*19+8 
Remainder = 8(!=0), again Apply Division lemma 

Divisor=8,Dividend=32,Quotient=32/8=4 
Euclid’s Div. Algo: 32=8*4+0 
Remainder = 0,We have to stop here 
HCF(616,32)=8 

Therefore, Maximum Number of Columns Will be 8 in which Both groups can march

Question 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.

Solution:

Let x be any positive integer, And y=3, 
As per Euclid’s Div algo. x=3q+r;q>=0 && 0 <= r < 3 => r=(0,1,2) 

If r=0: x=3q 
Squaring both sides: x2 =3*3*q2 
let assume m1=3q2 ,Here m1 will be any positive integer because q>=0. 
=>x2=3m1 
if r=1: x=3q+1 
Squaring both sides:x2 =(3q+1)2 
=>x2 =3(3q2 + 2q)+1 
let assume m2=3q2+2q 
=> x2 = 3m2 + 1 ,Here m2 will be any positive integer because q>=0. 
if r=2 : x=3q+2 
Squaring both sides:x2 =(3q+2)2 
=>x2 =3(3q2 + 4q+1)+1 
let assume m3=3q2+4q+1 
=> x2 = 3m3 + 1 ,Here m3 will be any positive integer because q>=0. 
Since m1,m2,m3 are positive integer ,therefore we can Conclude that x2=3m or 3m+1 ,Where m is some integer. 
This proved that the square of any positive integer(x) is either of the form 3m or 3m + 1 for some integer m

Question 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 x be any positive integer, and y=3 
As per Euclid’s Div algo. x=3q+r; q>=0 && 0 <= r < 3 => r=(0,1,2) 

if r=0: x=3q 
Cubing both sides: x3 =9*3*q3 
let assume m1=3q3 ,Here m1 will be any positive integer because q>=0. 
=>x2=9m1 
if r=1: x=3q+1 
Cubing both sides: x3 =(3q+1)3 
=>x3 =9(3q3+3q2+q)+1 
Let assume m2=3q3+3q2+q 
=> x3 = 9m2 + 1 ,Here m2 will be any positive integer because q>=0. 
if r=2 : x=3q+2 
Cubing both sides: x3 =(3q+2)3 
=>x3 =9(3q3 + 6q2+4q)+8 
let assume m3=3q3+6q2+4q 
=> x3 = 9m3 + 8 ,Here m3 will be any positive integer because q>=0. 
Since m1,m2,m3 are positive integer ,therefore we can Conclude that x3=9m,9m+1 or 3m+8, where m is some integer. 
This proved that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8. 

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