Question 1

About the number of pairs which have 16 as their H.C.F and 136 as their L.C.M. We can definitely say that?

Accepted Answer

H.C.F. of two numbers divides their L.C.M. exactly. Since 16 is not a factor of 136, it states that there does not exist any pair of numbers with H,C.F. 16 and L.C.M. 136.

Question 2

The least number, which when divided by 16, 18 and 21 leaves the remainder 3, 5 and 8 respectively, is?

Accepted Answer

Here (16 - 3) = 13, (18 - 5) = 13 and (21 - 8 ) = 13 So, required number = (L . C . M of 16, 18, 21) - 13 = (1008 - 13) = 995

Question 3

The greatest number of four digits which is divisible by each one of the numbers 12, 18 21 and 28 is?

Accepted Answer

Required number is divisible by the L . C . M of 12, 18, 21, 28 i.e 252. Now, greatest number of four digits = 9999 On dividing 9999 by 252, the remainder is 171 ∴ Required number = (9999 - 171) = 9828.

Question 4

The least number which when divided by 15, 27, 35 and 42, leaves in each case a remainder 7 is?

Accepted Answer

Required Number = (LCM of 15, 27, 35 and 42) + 7 = 1890 + 7 = 1897

Question 5

The measuring rods are 64 cm, 80 cm and 96 cm in length. The least length of cloth that can be measured exact number of times using any one of the above rods bis?

Accepted Answer

Required length = L. C . M of (64, 80, 96) cm = 960 cm = 9.60 m.

Question 6

The number of prime factors in the expression (6) 10 x (7) 17 x (11) 27 is?

Accepted Answer

Since 2, 3, 7, 11 are prime numbers and the given expression is 210 x 310 x 717 x 1127 So the number of prime factors in the given expression is (10 + 10 + 17 + 27 ) = 64.

Question 7

What least number must be subtracted from 1294 so that the remainder when divided by 9, 11, 13 will leave in each case the same remainder 6?

Accepted Answer

L. C .M of 9, 11, 13 is 1287 On dividing 1294 by 1287, the remainder is 7 . ∴ 1 must be subtracted from 1294, so that 1293 when divided by 9, 11, 13 leaves in each case the same remainder 6 .

Question 8

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?

Accepted Answer

L.C.M of 5, 6 , 7, 8 is 840 So, the 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

Question 9

The greatest number by which if 1657 and 2037 are divided the remainders will be 6 and 5 respectively, is?

Accepted Answer

Required number = H.C.F of (1657 - 6) and (2037 - 5) = H.C.F of 1651 and 2032 = 127

Question 10

Six bell commence tolling together and toll at intervals of 5, 10, 15, 20, and 30 s, respectively. In 60 min, how many times do they toll together?

Accepted Answer

LCM of 5, 10, 15, 20, 25 and 30 is 300. So, the bell will toll together after every 300s (5min). So, the number of times they toll together = 60/5 + 1=13

Question 11

If the HCF of two positive integers is 26. Then their LCM cannot be?

Accepted Answer

LCM of the number is always divisible by the HCF of the same numbers. So, 78, 104 and 234 are all divisible by 26. Whereas, 144 is not divisible by 26. Thus, 144 cannot be the LCM of the number whose HCF is 26.

Question 12

If the HCF of m and n is 1, then what are the HCF of m + n, m and HCF of m - n, n, respectively (where, m > n)?

Accepted Answer

The HCF of m and n is 1, so m and n are prime number. Let m = 7 and n = 5 ⇒ m + n = 12 HCF of(m + n)and m = HCF of 12 and 7 = 1 Similarly, HCF of (m - n) and n = 1

Question 13

The LCM and HCF of two number are 240 and 16, respectively,If the two numbers are in the ratio 3 : 5, the numbers are?

Accepted Answer

Suppose two numbers are 3N and 5N. Then, 3N x 5N = HCF x LCM ⇒ 15N2 = 16 x 240 ⇒ N2 = 256 ⇒ N = 16 So, the number are 3N = 3 x 16 = 48 and 5N = 5 x 16 = 80.

Question 14

What are the values of c when the HCF of x 3 + c x 2 - x + 2c and x 2 + cx - 2 over the rational is a linear polynomial?

Accepted Answer

Find HCF of x3 +c x2 -x + 2c and x2 + cx - 2 by long division method and get remainder ∴ Remainder = 2c - 2/c Since,remainder should be zero ∴ 2c2 - 2 = 0 ⇒ 2(c2 -1) = 0 ⇒ c= ± 1

Question 15

What is the greatest number that divides 263, 935 and 1383 leaving a remainder of 7 in each case?

Accepted Answer

The required greatest number is the HCF of (263 - 7, 935 - 7, 1383 - 7) i.e., HCF of 256, 928 and 1376 = 32

Question 16

What is the greatest number which divides 390, 480 and 620 so as to leave the same remainder in each case?

Accepted Answer

The HCF of [(480 - 390), (620 - 480), (620 - 390)] = HCF of 90, 140, 230 = 10

Question 17

Consider the number between 300 and 400 which when divided by 12, 18 and 36 leaves 4 as remainder in each case. what is the sum of the number?

Accepted Answer

LCM of (12,18, 36) = 36 ∴ A number greater than 300 on dividing by 36 leaves a remainder 4 = (36 x 9) + 4 = 328 Next numbe r = 328 + 36 = 364 ∴ Sum of number = 328 + 364 = 692

Question 18

Find the HCF of 132, 204 and 228.

Accepted Answer

132 = 2 x 2 x 3 x 11 240 = 2 x 2 x 2 x 2 x 3 x 5 228 = 2 x 2 x 3 x 19 Hence, HCF of 132, 204 and 228 = 2 x 2 x 3 = 12

Question 19

Find the HCF of 4/5 and 7/15?

Accepted Answer

We know that, HCF of fractions = (HCF of numerators) / (LCM of denominators) ∴ Required HCF = (HCF of 4 and 7) / (LCM of 5 and 15) =1/15

Question 20

Find the HCF of 1/2, 3/4, and 4/5?

Accepted Answer

Required HCF = (HCF of numerators) / (LCM of denominators) = (HCF of 1, 3 and 4) / (LCM of 2, 4, and 5) = 1/20

Problems on H.C.F and L.C.M Questions