Finding the Greatest 5-Digit Number Divisible by 15, 21, and 35
The problem asks for the largest number with five digits that can be evenly divided by 15, 21, and 35.
Step 1: Calculate the Least Common Multiple (LCM)
First, find the LCM of the given divisors: 15, 21, and 35. This LCM is the smallest number divisible by all three.
- Prime factorization of 15: $3 \times 5$
- Prime factorization of 21: $3 \times 7$
- Prime factorization of 35: $5 \times 7$
The LCM is the product of the highest powers of all prime factors involved:
LCM(15, 21, 35) = $3 \times 5 \times 7 = 105$.
Step 2: Identify the Largest 5-Digit Number
The largest possible number with five digits is 99999.
Step 3: Find the Divisible Number
To find the greatest 5-digit number divisible by 105, divide the largest 5-digit number (99999) by the LCM (105) and take the integer part of the quotient.
$ \frac{99999}{105} \approx 952.37 $
The integer quotient is 952.
Now, multiply this quotient by the LCM to get the desired greatest 5-digit number:
$ 952 \times 105 = 99960 $
Conclusion
The greatest number of five digits that is divisible by 15, 21, and 35 is 99960.
Similar Questions
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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?
