Real Numbers Exercise – 1.1 NCRT Download PDF || UP Board
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 225
(i) 135 and 225
Solution:- 225>135
Let a = 225, b =135
From
Euclid's division algorithm,
a = b ×q +
r, 0 ≤ r < b
225 = 135×1 +
90 (r≠0, When r is not equal to zero, the theorem is used again.)
135 = 90×1 +
45 (r≠0)
90 = 45×2 + 0 (r=0, The solution is terminated when r
is equal to zero and the value of b is HCF.)
HCF = 45
(ii)
196 and 38220
Solution:- 38220>196
Let, a =
38220, b=196
From
Euclid's division algorithm,
a = b×q + r,
0≤ r < b
38220 =
196×195 + 0 (r=0, The solution is terminated when r is equal to zero and the
value of b is HCF.)
HCF = 196
(iii) 867 and
225
Solution:- 867>225
Let a = 867,
b= 225
From
Euclid's division algorithm,
a = b×q + r,
0≤ r < b
867 = 225×3
+ 102 (r≠0)
225 = 102×2
+ 51 (r≠0)
102 = 51×2 +
0 (r=0, The solution is terminated when r is equal to zero and the value of b
is HCF.)
HCF = 51
2. Show that
any positive odd integer is of the form 6q+1, 6q+3 or 6q+5, where q is some integer.
Solution:-
we have to
show _ a = 6q+1, 6q+3 या 6q+5
Let a be any
positive odd integer and b = 6, then by Euclid's division algorithm,
a = 6q+r, r= 0,1,2,3,4,5 (0≤ r <6, The divisor cannot be greater than
nor equal to the remainder)
Put the
value of 'r',
If r = 2, so a = 6q
Similarly, Putting
r = 1,2,3,4,5 will give a = 6q+1, 6q+2, 6q+3, 6q+4, 6q+5 respectively.
Hence, a
positive odd integer will be of the form 6q+1, 6q+3 or 6q+5.
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:
The maximum number of columns will
be equal to HCF(616, 32).
Since, 616 > 32
Then, from
Euclid's algorithm,
616 = 32 ×
19 + 8 ( r ≠ 0, So 32 as the new dividend and 8 as the new divisor )
32 = 8 × 4 +
0 (r =0, then
the value of b is HCF)
b = 8
HCF(616, 32)
= 8
So the
maximum number of columns will be 8.
4 . Use
Euclid’s division lemma to show that the square of any positive integer is
either of form 3m or 3m + 1 for some integer m.
Solution:
Let ‘a’ be
any positive integer and b = 3
Then, from
Euclid's algorithm,
a = bq + r for some integers q ≥ 0 and r = 0,1,2 where 0 ≤ r < 3
therefore, a
= 3q, a = 3q+1, a = 3q+2
According to the question, on squaring all the equations on
both sides,
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
Then, from
Euclid's algorithm,
a = bq + r for some integers q ≥ 0 and r = 0,1,2 where 0 ≤ r < 3
When r = 0,
a = 3q
Hence, the cube of a positive integer will be of the form 9m, 9m+1 or 9m+8.
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