RD Chapter 1 Real Numbers Ex MCQS Solutions
Question - 31 : - The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is
(a) 10
(b) 100
(c) 504
(d) 2520
Answer - 31 : -
(d) Factors of 1 to 10 numbers
1 = 1
2 = 1 x 2
3 = 1 x 3
4 = 1 x 2 x 2
5 = 1 x 5
6 = 1 x 2 x 3
7 = 1 x 7
8 = 1 x 2 x 2 x 2
9 = 1 x 3 x 3
10 = 1 x 2 x 5
LCM of number 1 to 10 = LCM (1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
= 1 x 2 x 2 x 2 x 3 x 3 x 5 x 7 = 2520
Question - 32 : - The largest number which divides 70 and 125, leaving remainders 5 and 8, respectively, is
(a) 13
(b) 65
(c) 875
(d) 1750
Answer - 32 : -
(a) Since, 5 and 8 are the remainders of 70 and 125, respectively. Thus, after subtracting these remainders from the numbers, we have the numbers 65 = (70 тАУ 5), 117 = (125 тАУ 8), which is divisible by the required number.
Now, required number = HCF of 65, 117
[For the largest number]
For this, 117 = 65 x 1 + 52 [Dividend = divisor x quotient + remainder]
=> 65 = 52 x 1 + 13
=> 52 = 13 x 4 + 0
HCF = 13
Hence, 13 is the largest number which divides 70 and 125, leaving remainders 5 and 8.
Question - 33 : - If the HCF of 65 and 117 is expressible in the form 65m тАУ 117, then the value of m is
(a) 4
(b) 2
(c) 1
(d) 3
Answer - 33 : -
(b) By EuclidтАЩs division algorithm,
b = aq + r, 0 тЙд r < a [dividend = divisor x quotient + remainder]
=> 117 = 65 x 1 + 52
=> 65 = 52 x 1 + 13
=> 52 = 13 x 4 + 0
HCF (65, 117)= 13 тАж(i)
Also, given that HCF (65, 117) = 65m тАУ 117 тАж..(ii)
From equations (i) and (ii),
65m тАУ 117 = 13
=> 65m = 130
=> m = 2
Question - 34 : - The decimal expansion of the rational number 14587/1250 will terminate after:
(a) one decimal place
(b) two decimal places
(c) three decimal places
(d) four decimal places
Answer - 34 : - (d)
Hence, given rational number will terminate after four decimal places.
Question - 35 : - EuclidтАЩs division lemma states that for two positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy
(a) 1 < r < b
(b) 0 < r тЙд b
(c) 0 тЙд r < b
(d) 0 < r < b
Answer - 35 : -
(c)┬аAccordingto EuclidтАЩs Division lemma, for a positive pair of integers there exists uniqueintegers q and r, such that
a = bq + r, where 0 тЙд r < b