Question 1

H.C.F of 4 x 27 x 3125, 8 x 9 x 25 x 7 and 16 x 81 x 5 x 11 x 49 is

Accepted Answer

To find the HCF (Highest Common Factor) of the given numbers, we begin by breaking down each number into its prime factorization. Here are the steps: Prime factorization of 4 × 27 × 3125: 4 = 2² 27 = 3³ 3125 = 5⁵ So, the prime factorization of 4 × 27 × 3125 is 2² × 3³ × 5⁵. Prime factorization of 8 × 9 × 25 × 7: 8 = 2³ 9 = 3² 25 = 5² 7 = 7¹ So, the prime factorization of 8 × 9 × 25 × 7 is 2³ × 3² × 5² × 7¹. Prime factorization of 16 × 81 × 5 × 11 × 49: 16 = 2⁴ 81 = 3⁴ 5 = 5¹ 11 = 11¹ 49 = 7² So, the prime factorization of 16 × 81 × 5 × 11 × 49 is 2⁴ × 3⁴ × 5¹ × 7² × 11¹. Find the common prime factors and their lowest powers: For 2, the lowest power is 2². For 3, the lowest power is 3². For 5, the lowest power is 5¹. For 7, the lowest power is 7⁰ (since 7 is absent in the first number). For 11, the lowest power is 11⁰ (since 11 is absent in the first two numbers). Calculate the HCF: HCF = 2² × 3² × 5¹ = 4 × 9 × 5 = 180. Final answer: The HCF of the given numbers is 180.

Question 2

A room is 6 meters 24 centimeters in length and 4 meters 32 centimeters in Width. Find the least number of square tiles of equal size required to cover the entire floor of the room.

Accepted Answer

Let us calculate both the length and width of the room in centimeters. Length = 6 meters and 24 centimeters = 624 cm width = 4 meters and 32 centimeters = 432 cm As we want the least number of square tiles required, it means the length of each square tile should be as large as possible.Further,the length of each square tile should be a factor of both the length and width of the room. Hence, the length of each square tile will be equal to the HCF of the length and width of the room = HCF of 624 and 432 = 48 Thus, the number of square tiles required = (624 x 432 ) / (48 x 48) = 13 x 9 = 117

Question 3

The largest measuring cylinder that can accurately fill 3 tanks of capacity 98, 182 and 266 litres each, is of capacity ?

Accepted Answer

To know the the measuring cylinder that can fill all the given capacities , they must be divisible by the required number. 98,182,266 all are divisible by 14 So 14 litres is the largest cylinder that can fill all the given cylinders. (or) The other method is take HCF of all given capacities i.e 98, 182 and 266.

Question 4

If N is the greatest number that will divide 1305, 4665 and 6905, leaving the same remainder in each case. What is the sum of the digits of N ?

Accepted Answer

N = H.C.F. of (4665 - 1305), (6905 - 4665) and (6905 - 1305) = H.C.F. of 3360, 2240 and 5600 = 1120. Sum of digits in N = ( 1 + 1 + 2 + 0 ) = 4

Question 5

The G.C.D of 1.08, 0.36 and 0.9 is

Accepted Answer

Given numbers are 1.08 , 0.36 and 0.90 H.C.F of 108, 36 and 90 is 18 [ ∵ G.C.D is nothing but H.C.F] Therefore, H.C.F of given numbers = 0.18

Question 6

Find the greatest number that will divide 964, 1238 and 1400 leaving remainder of 41,31 and 51 respectively ?

Accepted Answer

To find the greatest number which divides the numbers 964, 1238 and 1400 leaving the remainders 41, 31 and 51 is nothing but the HCF of (964 - 41), (1238 - 31), (1400 - 51). Therefore, HCF of 923, 1207 and 1349 is 71.

Question 7

Find the least common multiple of 15 and 12?

Accepted Answer

The least common multiple of 12 and 15 is 12 = 2 x 2 x 3 15 = 3 x 5 LCM of 12 and 15 is 2 x 2 x 3 x 5 = 60.

Question 8

Three numbers are in the ratio of 3:4:5 and their L.C.M is 3600.Their HCF is:

Accepted Answer

Let the numbers be 3x, 4x, 5x. Then, their L.C.M = 60x. So, 60x=3600 or x=60. Therefore, The numbers are (3 x 60), (4 x 60), (5 x 60). Hence,required H.C.F=60

Question 9

In a palace, three different types of coins are there namely gold, silver and bronze. The number of gold, silver and bronze coins is 18000, 9600 and 3600 respectively. Find the minimum number of rooms required if in each room should give the same nu…

Accepted Answer

Gold coins = 18000 , Silver coins = 9600 , Bronze coins = 3600 Find a number which exactly divide all these numbers That is HCF of 18000, 9600& 3600 All the value has 00 at end so the factor will also have 00. HCF for 180, 96 & 36. Factors of 180 = 3 x 3 x 5 x 2 x 2 96 = 2 x 2 x 2 x 2 x 2 x 3 36 = 2 x 2 x 3 x 3 Common factors are 2x2×3=12 Therefore, Actual HCF is 1200 Gold Coins 18000/1200 will be in 15 rooms Silver Coins 9600/1200 will be in 8 rooms Bronze Coins 3600/1200 will be in 3 rooms Total rooms will be (15+8+3) = 26 rooms.

Question 10

Three different containers contain 1365 litres, 1560 litres and 1755 litres of mixtures of milk and water respectively. What biggest measure that can measure all the different quantities exactly ?

Accepted Answer

From the given data, containers contain mixtures of milk and water as 1365 lit, 1560 lit and 1755 lit. Biggest measure that can measure all these different quantities exactly can be given by HCF of (1365, 1560, 1755) That can be found as HCF of 1365, 1755 = 195 Now, HCF of 195 and 1560 = 195 Hence, biggest measure that can measure all the given different quantities exactly is 195 lit.

Question 11

A natural number, when divided by 9, 10, 12 or 15, leaves a remainder of 3 in each case. What is the smallest of all such numbers?

Accepted Answer

LCM of 9,10,12,15 is 180 Smallest natural number to get remainder of 3 is (180 + 3) = 183

Question 12

A student can divide her books into groups of 5, 9 and 13. what is the smallest possible number of the books ?

Accepted Answer

The smallest possible number of books = L.C.M of 5, 9 and 13. Therefore, L.C.M of 5,9 and 13 is = 5 x 9 x 13 = 585.

HCF and LCM Questions