Practice MCQ Questions and Answer on Problems on H.C.F and L.C.M
81.
The greatest 4-digit number exactly divisible by 10, 15, 20 is = ?
(A) 9990
(B) 9960
(C) 9980
(D) 9995
Solution:
LCM of (10, 15, 20) ⇒ 5 × 2 × 3 × 2 = 60 ⇒ Largest 4-digit number = 9999 divide 9999 by LCM of given number ⇒ We get remainder = 39 ⇒ So, to divide completely subtract it from (9999 - 39) = 9960 ∴ 9960 is the largest four digit number which is completely divided by the given numbers.
82.
The LCM and ratio of four numbers are 630 and 2 : 3 : 5 : 7 respectively. The difference between the greatest and least numbers is = ?
(A) 6
(B) 14
(C) 15
(D) 21
Solution:
Let the numbers be 2x, 3x, 5x and 7x respectively Then their LCM = (2 × 3 × 5 × 7)x = 210x.[∴ 2, 3, 5, 7 are prime numbers] So, 210x = 630 or x = 3 ∴ The numbers are 6, 9, 15 and 21 Required difference = 21 - 6 = 15
83.
If HCF of p and q is x and q = xy, then the LCM of p and q is = ?
The least number which when divided by 5, 6, 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder, is = ?
(A) 1677
(B) 1683
(C) 2523
(D) 3363
Solution:
LCM of 5, 6, 7, 8 = 840 ∴ Required number is of the from 840k + 3 Least value of k for which (840k + 3) is divisible by 9 is k = 2 ∴ Required number = (840 × 2 + 3) = 1683
85.
Let N be the greatest number that will divide 1305, 4665 and 6905 leaving the same remainder in each case. Then sum of the digits in N is = ?
(A) 4
(B) 5
(C) 6
(D) 8
Solution:
N = HCF of (4665 - 1305) (6905 - 4665) and (6905 - 1305) = HCF of 3360, 2240 and 5600 = 1120 Sum of digits in N = (1 + 1 + 2 + 0) = 4
86.
The least number which when divided by 5, 6 , 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder, is:
(A) 1677
(B) 1683
(C) 2523
(D) 3363
Solution:
L.C.M. of 5, 6, 7, 8 = 840. ∴ Required number is of the form 840k + 3 Least value of k for which (840k + 3) is divisible by 9 is k = 2 ∴ Required number = (840 x 2 + 3) = 1683
87.
A gardener has to plant trees in rows containing equal number of trees. If he plants in rows of 6, 8, 10 or 12 then five trees are left unplanted. But if he plants in rows of 13 trees each, then no tree is left. What is the number of trees that the gardener plants ?
(A) 485
(B) 725
(C) 845
(D) None of these
Solution:
LCM of 6, 8, 10, 12 = 120 ∴ Required number is of the from 120k + 5 Least value of k for which (120k + 5) is divisible by 13 is k = 7 ∴ Required number = (120 × 7 + 5) = 845
88.
What will be the least number which when doubled will be exactly divisible by 12, 18, 21 and 30 ?
The product of two numbers is 2008 and their HCF is 13. The number of such pairs is = ?
(A) 1
(B) 2
(C) 3
(D) 4
Solution:
Let the numbers be 13a and 13b Then, 13a × 13b = 2028 ab = 12 Now, co - primes with product 12 are (1, 12) and (3, 4) So, the required numbers are (13 × 1, 13 × 12) and (13 × 3, 13 × 4) Clearly, there are 2 such pairs
90.
The least number which when divided by 6, 9, 12, 15, 18 leaves the same remainder 2 in each case is = ?
(A) 180
(B) 176
(C) 182
(D) 178
Solution:
LCM of 6, 9, 12, 15, 18 is = 180 If 180 is divided by these given number remainder will be 0 ⇒ To leave the same remainder 2 ⇒ The number will be = 180 + 2 = 182