NCERT Math solutions Chapter 1 Real Numbers Exercise 1.1
1.
Use Euclid's division algorithm to find the HCF of:
(i) 135 and 225 (ii) 196 and 38220 (iii)
867 and 255
(i)
135 and 225
To
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
c=dq+r,0≤r<d
Here, c=225,d=135
225=135×1+90
Remainder
is not equal to 0. Therefore, we apply the same process again on 135 and 90
135=90×1+45
Remainder
is not equal to 0 again. Therefore, we apply same process again on 90 and 45.
90=45×2+0
Remainder
is equal to 0.
Therefore, HCF of 135 and 225 is equal to 45 which is equal to value
of d in the last step.
(ii)
196 and 38220
To
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
c=dq+r,0≤r<d
Here, c=38220,d=196
38220=196×195+0
Remainder
is equal to 0.
Therefore, HCF of 196 and 38220 is equal to 196 which is equal to value
of d in the last step.
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:
Let a be
any positive odd integer and b=6.
We can apply Euclid's division algorithm on a and b=6.
We know that value of 0≤r<6 .
Therefore, all possible values of a are:
a=6q
a=6q+1
a=6q+2
a=6q+3
a=6q+4
a=6q+5
We can ignore 6q, 6q+2 and 6q+4 because they are divisible by 2 which
means they are not positive odd integers.
Therefore, we are just left with 6q+1, 6q+3 and 6q+5.
Therefore, any positive odd
integer is of the form (6q+1) or (6q+3)or (6q+5).
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
Here, c=616,d=32
Remainder is not equal to 0.
Therefore, we apply the same process again on 32 and 8.
Remainder is equal to 0.
Therefore,
HCF of 616 and 32 is equal to 8 which
is equal to value of d in the last step.
It
means that they can march in maximum of 8 columns.
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 3q,3q+1,3q+2 . 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:
Now lets square each one of them
one by one.
(i) (3q)2= 9q 2
(ii) (3q+1)2 = 9q 2 +6q+1=3(3q 2 +2q)+1
Let m=3q 2+2q be some integer, we get
(iii) (3q+2) 2 =9q 2 +4+12q=9q 2 +12q+3+1=3(3q 2 +4q+1)+1
Let m=(3q 2 +4q+1) be some integer, we get
Hence, square of any positive
integer is either of the form 3m or 3m+1 for some integer m.
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:
Now, we will find cube of
each one of them.
(i)
(ii)
Let m=(3q+3q 2 +q) be any integer then we get,
(iii)
Let m=(3q3 +6q 2 +4q) be any integer, then we get
Therefore,
it is proved that cube of any positive number is of the form 9m, 9m+1 or 9m+8.
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