Finding Numbers Divisible by 21, 35, and 63 Less Than 10000
To find the count of numbers less than 10000 that are divisible by 21, 35, and 63, we first need to find the Least Common Multiple (LCM) of these three numbers. A number divisible by all three must be a multiple of their LCM.
Calculating the LCM
First, find the prime factorization of each number:
- $21 = 3 \times 7$
- $35 = 5 \times 7$
- $63 = 3^2 \times 7$
The LCM is found by taking the highest power of each prime factor present in any of the factorizations:
LCM$(21, 35, 63) = 3^2 \times 5 \times 7 = 9 \times 5 \times 7 = 315$
Counting Multiples Less Than 10000
We need to find how many multiples of 315 are less than 10000. Let the number of multiples be $n$. We are looking for the largest integer $n$ such that:
$ n \times 315 < 10000 $To find $n$, we divide 10000 by 315:
$ n < \frac{10000}{315} $ $ n < 31.746... $The largest whole number less than 31.746... is 31. Therefore, there are 31 multiples of 315 that are less than 10000.
Conclusion
There are 31 numbers less than 10000 that are exactly divisible by 21, 35, and 63.
Similar Questions
- 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 $12 \times 15, 15 \times 21, 21 \times 12$
- 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 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 $\sqrt[2]{144}$ ?
- The HCF of $28\ p^5q^2$ and $70\ p^3q^4$ is:
- If the HCF of 85 and 153 is expressible in the form of $85x - 153$, then the value of x is:
- What is the greatest number which divides 13, 26 and 39 and gives the remainders as 1, 2 and 3 respectively?
- The LCM of x and y is 2079 and their HCF is 27. If x = 189, what will be the value of y?
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?
