Find the greatest number of three digits that can be exactly divided by 75,45 and 60
Lcm of 75 45 60=900 that is exactly divided.
Math
To find the greatest number of three digits that can be exactly divided by 75, 45, and 60, we need to find the least common multiple (LCM) of these three numbers.
Step 1: Prime Factorization
First, we need to find the prime factorization of each number:
75 = 3 × 5^2
45 = 3^2 × 5
60 = 2^2 × 3 × 5
Step 2: LCM Calculation
To find the LCM, we take the highest power of each prime factor that appears in the prime factorization of any of the numbers.
In this case, the LCM would be:
2^2 × 3^2 × 5^2 = 900
Step 3: Three-digit Number multiple of 900
To find the greatest three-digit number that is a multiple of 900, we start with the largest three-digit number (999) and find the largest multiple of 900 that is less than or equal to 999.
By dividing 999 by 900, we find that the quotient is 1 with a remainder of 99.
Subtracting the remainder (99) from 999 gives us the greatest three-digit number that is evenly divisible by 900, which is 900.
Therefore, the greatest number of three digits that can be exactly divided by 75, 45, and 60 is 900.