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26. The least number that is divisible by all the numbers from 1 to 8 (both inclusive) is
(a) 840
(b) 2520
(c) 8
(d) 420
Answer
Answer: a
29. The product of two consecutive natural numbers is always:
(a) prime number
(b) even number
(c) odd number
(d) even or odd
Answer
Answer: b
30. If the HCF of 408 and 1032 is expressible in the form 1032 x 2 + 408xp, then the value of p is
(a) 5
(b) -5
(c) 4
(d) -4
Answer/ Explanation
Answer: b
Explaination:(b); [Hint. HCF of 408 and 1032 is 24, .-. 1032 x 2 + 408 x (-5)]
31. The number in the form of 4p + 3, where p is a whole number, will always be
(a) even
(b) odd
(c) even or odd
(d) multiple of 3
Answer
Answer: b
32. When a number is divided by 7, its remainder is always:
(a) greater than 7
(b) at least 7
(c) less than 7
(d) at most 7
Answer
Answer: c
33. (6 + 5 √3) – (4 – 3 √3) is
(a) a rational number
(b) an irrational number
(c) a natural number
(d) an integer
Answer
Answer: b
34. If HCF (16, y) = 8 and LCM (16, y) = 48, then the value of y is
(a) 24
(b) 16
(c) 8
(d) 48
Answer
Answer: a
35. According to the fundamental theorem of arith-metic, if T (a prime number) divides b2, b > 0, then
(a) T divides b
(b) b divides T
(c) T2 divides b2
(d) b2 divides T2
37. If LCM (77, 99) = 693, then HCF (77, 99) is
(a) 11
(b) 7
(c) 9
(d) 22
Answer
Answer: a
38. Euclid’s division lemma states that for two positive integers a and b, there exist unique integer q and r such that a = bq + r, where r must satisfy
(a) a < r < b
(b) 0 < r ≤ b
(c) 1 < r < b
(d) 0 ≤ r < b
Answer
Answer: d
39. For positive integers a and 3, there exist unique integers q and r such that a = 3q + r, where r must satisfy:
(a) 0 < r < 3
(b) 1 < r < 3
(c) 0 < r < 3
(d) 0 < r < 3
Answer
Answer: a
40. Find the greatest number of 5 digits, that will give us remainder of 5 when divided by 8 and 9 respectively.
(a) 99921
(b) 99931
(c) 99941
(d) 99951
Answer/ Explanation
Answer: c
Explaination: The greatest number will be multiple of LCM (8, 9)
LCM of 8 and 9 = 72
On verification we find that 99941 when divided by 72 leaves remainder 5.
41. For some integers p and 5, there exist unique integers q and r such that p – 5q + r. Possible values of r are
(a) 0 or 1
(b) 0, 1 or 2
(c) 0, 1, 2 or 3
(d) 0, 1, 2, 3 or 4
Answer/ Explanation
Answer: d
Explaination: According to Euclids division lemma, p – 5q + r, where 0 < r < 5
r = 0,1,2, 3, 4
So, possible values of r are 0, 1,2, 3 or 4
43. If two positive integers p and q can be expressed as p = ab² and q = c3b; where a, b being prime numbers, then LCM (p, q) is equal to
(a) ab
(b) crb²
(c) a3b²
(d) c²b3
Answer/ Explanation
Answer: c
Explaination:LCM (p, q) = a3b²
44. The ratio between the LCM and HCF of 5, 15, 20 is:
(a) 9 : 1
(b) 4:3
(c) 11:1
(d) 12:1
Answer/ Explanation
Answer: d
Explaination:
5, 15 = 5×3, 20 = 2x2x5
LCM(5, 15, 20) = 5x3x2x2 = 60
HCF(5, 15, 20) = 5
45. Two alarm clocks ring their alarms at regular intervals of 50 seconds and 48 seconds. If they first beep together at 12 noon, at what time will they beep again for the first time ?
(a) 12.20 pm
(b) 12.12 pm
(c) 12.11pm
(d) none of these
Answer/ Explanation
Answer: d
Explaination:
LCM of 50 and 48 = 1200
1200 sec = 20 min
Hence at 12.20 pm they will beep again for the first time.
46. If A = 2n + 13, B = n + 7, where n is a
natural number, then HCF of A and B is:
(a) 2
(b) 1
(c) 3
(d) 4
Answer/ Explanation
Answer: b
Explaination:
Taking different values of n we find
that A and B are coprime.
HCF = 1
47. There are 576 boys and 448 girls in a school that are to be divided into equal sections of either boys or girls alone. The total number of sections thus formed are:
(a) 22
(b) 16
(c) 36
(d) 21
Answer/ Explanation
Answer: b
Explaination:
HCF of 576 and 448 = 64
Number of sections =
= 9 + 7 = 16
48. The HCF of 2472, 1284 and a third number N is 12. If their LCM is 23 x 32 x 5 x 103 x 107, then the number Nis :
(a) 22 x 32 x 7
(b) 22 x 33 x 103
(c) 22 x 32 x 5
(d) 24 x 32 x ll
Answer/ Explanation
Answer: c
Explaination:
2472 = 23x3x103
1284 = 2²x3x107
LCM = 23x3²x5x103x107
N = 2²x3²x5 = 180
49. Two natural numbers whose difference is 66 and the least common multiple is 360, are:
(a) 120 and 54
(b) 90 and 24
(c) 180 and 114
(d) 130 and 64
Answer/ Explanation
Answer: b
Explaination:
Difference of 90 and 24 = 66
and LCM of 90 and 24 = 360
∴ Numbers are 90 and 24.
50. HCF of 52 x 32 and 35 x 53 is:
(a) 53 x 35
(b) 5 x 33
(c) 53 x 32
(d) 52 x 32
Answer/ Explanation
Answer: d
Explaination:
HCF of 5²x3² and 35x53
= 5²x3²