Finding the Smallest Five-Digit Number Divisible by Given Numbers
The goal is to identify the smallest integer that has five digits and is exactly divisible by 12, 16, 24, and 28.
Step 1: Calculate the Least Common Multiple (LCM)
To find a number divisible by all the given numbers (12, 16, 24, 28), we first calculate their LCM. This is the smallest positive number that is a multiple of all of them.
Find the prime factorization of each number:
- $12 = 2^2 \times 3$
- $16 = 2^4$
- $24 = 2^3 \times 3$
- $28 = 2^2 \times 7$
The LCM is the product of the highest powers of all prime factors involved:
LCM$(12, 16, 24, 28) = 2^4 \times 3^1 \times 7^1 = 16 \times 3 \times 7 = 336$.
Step 2: Find the Smallest Five-Digit Multiple of the LCM
The smallest five-digit number is 10000.
To find the smallest five-digit number divisible by 336, we divide 10000 by 336 and find the next whole number multiple.
$ \frac{10000}{336} \approx 29.76 $
The next integer after 29.76 is 30. This means the 30th multiple of 336 is the smallest multiple that is a five-digit number.
Calculate the 30th multiple:
$ 30 \times 336 = 10080 $
Result
The smallest five-digit number divisible by 12, 16, 24, and 28 is 10080.
Similar Questions
- The H.C.F. and the L.C.M. of two numbers are 5 and 495, respectively. If one of the numbers is 55, find the other one.
- The HCF of two numbers is 16 and their LCM is 240. If one number is 48, find the other number.
- If the LCM of $20x^3y^2$ and $10x^4y^4$ is $20x^4y^4$, find the HCF.
- There are four table clocks. They ring every 10 min, 15 min, 20 min and 25 min respectively. If they all ring together at 10 a.m., then at what time will they ring together again?
- Find the HCF of $12 \times 15, 15 \times 21, 21 \times 12$
- How many numbers less than 10000 are there which are exactly divisible by 21, 35 and 63?
- A, B and C begin together to move around a circular stadium and they complete their revolutions in 42 s, 63 s and 84 s respectively. After how much time will they come together at the starting point?
- The HCF and the LCM of two numbers are 17 and 1224, respectively. If one of the numbers is 136, find the other one.
- The LCM of two numbers is 721, and the numbers are in the ratio of 1 : 7. What is the sum of the numbers?
What is the LCM of $\sqrt[2]{169}$, $\sqrt[3]{27}$, $\sqrt[3]{64}$ and $\sqrt[2]{144}$ ?
Important Questions from LCM and HCF
The HCF and LCM of two numbers are 12 and 72, respectively. If the ratio of the two numbers is 2 ∶ 3, then the larger of the two numbers is:
Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.
Joseph visits the club on every 5 th day, Harsh visits on every 24 th day, while Sumit visits on every 9 th day. If all three of them met at the club on a Sunday, then on which day will all three of them meet again?
What is the least number which when divided by 12,20 and 24 leaves in each case a remainder of 8?
Which of the following is a pair of co-primes?
