The Muscle Gym has bought 63 treadmills and 108 elliptical machines. The gym divides them into several identical sets of treadmills and elliptical machines for its branches located throughout the city, with no exercise equipment left over. What is the greatest number of branches the gym can have in the city?
Answers
Given: Number of Treadmills = 63
Number of Elliptical machines = 108
Step 1 - What will represent the greatest number of branches?
Let the greatest number of branches in the city be 'n'.
Let the number of treadmills sent to each branch be 't' and the number of ellipticals sent to each branch be 'e'.
63 treadmills were distributed to 'n' branches with each branch getting 't' treadmills.
i.e., nt = 63
108 elliptical machines were distributed to 'n' branches with each branch getting 'e' elliptical machines.
i.e., ne = 108.
So, n is a factor of 63 and 108.
'n' is the greatest number of branches the gym can have in the city. So, 'n' is the HCF of 63 and 108.
Step 2 - Find HCF of 63 and 108
Step 1: Prime factorise 63 and 108.
63 = 32 × 7
108 = 22 × 33
Step 2: List down common primes.
3 is the only prime common to both the numbers.
Step 3: HCF is the product of lowest power of common primes in the 2 numbers.
Lowest power of 3 between 63 and 108 is 32.
Therefore, HCF = 9
The greatest number of branches the gym can have in the city is 9.
Step-by-step explanation:
Prime factorise 63 and 108.
Prime factorise 63 and 108.63 = 32 × 7
Prime factorise 63 and 108.63 = 32 × 7108 = 22 × 33
Prime factorise 63 and 108.63 = 32 × 7108 = 22 × 33Step 2: List down common primes.
Prime factorise 63 and 108.63 = 32 × 7108 = 22 × 33Step 2: List down common primes.3 is the only prime common to both the numbers.
Prime factorise 63 and 108.63 = 32 × 7108 = 22 × 33Step 2: List down common primes.3 is the only prime common to both the numbers.Step 3: HCF is the product of lowest power of common primes in the 2 numbers.
Prime factorise 63 and 108.63 = 32 × 7108 = 22 × 33Step 2: List down common primes.3 is the only prime common to both the numbers.Step 3: HCF is the product of lowest power of common primes in the 2 numbers.Lowest power of 3 between 63 and 108 is 32.
Prime factorise 63 and 108.63 = 32 × 7108 = 22 × 33Step 2: List down common primes.3 is the only prime common to both the numbers.Step 3: HCF is the product of lowest power of common primes in the 2 numbers.Lowest power of 3 between 63 and 108 is 32.Therefore, HCF = 9
Prime factorise 63 and 108.63 = 32 × 7108 = 22 × 33Step 2: List down common primes.3 is the only prime common to both the numbers.Step 3: HCF is the product of lowest power of common primes in the 2 numbers.Lowest power of 3 between 63 and 108 is 32.Therefore, HCF = 9The greatest number of branches the gym can have in the city is 9.