The gcf(a,b) = 495 and lcm( a,b) =31,185 Find possible values of a and b if a is divisible by 35 and b is divisible by 81.
This is a problem that requires the understanding of the relationship between LCM and GCF.
Take the example of 45 and 63:
45=9*5
63=9*7
GCF=9
LCM=9*5*7
So you see that GCF*LCM equals the products of the two numbers.
Conversely, LCM/GCF is the product of factors NOT common to both (shown in bold in the above example).
To find numbers a,b which have given LCM and GCF, divide M=LCM/GCF and distribute factors of M to the GCF, for example,
LCM/GCF=9*7*5/9=7*5
Multiplying the GCF by each of the factors gives the original numbers:
9*7=63
9*5=45.
For the given case,
LCM=31185
GCF=495
LCM/GCF=63=7*9
So multiplying the GCF each by 7 and 9 will give the numbers a and b.
GCF=9*
find a pair of numbers for each set of conditions.the LCM is 35.the GCF is 7.
What is the greatest common factor of 23, 40, and 43?
find a pair of numbers that matches the descripion . the LCM of two numbers is 42 and their sum is 20
To find the possible values of a and b given the conditions, we can use the relationship between the greatest common factor (GCF) and the least common multiple (LCM).
We are given that the GCF of a and b is 495, and the LCM is 31,185.
We know that the GCF is the largest number that divides both a and b, while the LCM is the smallest multiple that is divisible by both a and b.
We also know that if two numbers are divisible by another number, their GCF must also be divisible by that number. Therefore, if a is divisible by 35 and b is divisible by 81, their GCF must also be divisible by both 35 and 81.
Let's break down the GCF and LCM into prime factors:
495 = 3^2 * 5 * 11
31,185 = 3^4 * 5 * 7
To find the GCF, we take the product of the common prime factors raised to the lowest exponent between the factorizations of a and b:
GCF = 3^2 * 5 = 45
Since the GCF (45) is divisible by 35, a must also be divisible by 35.
Now, let's find the LCM using the prime factorization:
LCM = 3^4 * 5 * 7 = 31,185
Since the LCM (31,185) is divisible by 81, b must also be divisible by 81.
Therefore, the possible values of a and b are any multiples of 35 and 81, respectively. Specifically, a must be a multiple of 35, and b must be a multiple of 81.