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.
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Important Questions from LCM and HCF
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Find the least number which when divided by 12, 18, 24 and 30 leaves 4 as remainder in each case, but when divided by 7 leaves no remainder.
Calculate the HCF of \(\frac{12}{5}\) , \(\frac{14}{15}\) and \(\frac{16}{17}\) .
Three numbers are in the proportion of 3 : 8 : 15 and their LCM is 8280. What is their HCF?
