Solved on Jan 07, 2024
Find two numbers whose lowest common multiple is 90 and highest common factor is 6.
STEP 1
Assumptions
1. The lowest common multiple (LCM) of the two numbers is 90.
2. The highest common factor (HCF) or greatest common divisor (GCD) of the two numbers is 6.
3. We need to find two numbers that satisfy these conditions.
STEP 2
Understand the relationship between the LCM and the HCF of two numbers. The product of the LCM and HCF of two numbers is equal to the product of the numbers themselves.
STEP 3
Plug in the given values for the LCM and HCF to find the product of the two numbers.
STEP 4
Calculate the product of the two numbers.
STEP 5
Now we need to find two factors of 540 that also have 6 as their highest common factor. To do this, we can list the factors of 540 and then find pairs that have 6 as their GCD.
STEP 6
List the factors of 540. The factors are all the numbers that can divide 540 without leaving a remainder. We can start with 1 and 540 and find other pairs of factors by dividing 540 by numbers starting from 2 upwards.
STEP 7
The factors of 540 are: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, 270, 540.
STEP 8
Now, we need to find pairs of these factors that have 6 as their highest common factor. We can do this by checking the GCD of different pairs.
STEP 9
Since the HCF is 6, we know that both numbers must be multiples of 6. Let's list the multiples of 6 that are factors of 540: 6, 12, 18, 36, 54, 90, 108, 180, 270.
STEP 10
We can now try to pair these factors to see which pairs have an LCM of 90. Remember that for two numbers to have an LCM of 90, one of the numbers must be a multiple of 90, and the other must be a factor of 90 that is not a multiple of a smaller number that is also a factor of 90.
STEP 11
We can start by considering 90 as one of the numbers since it is the LCM. The other number must be a factor of 540 and a multiple of 6, and when paired with 90, it must give an HCF of 6.
STEP 12
Pair 90 with each of the multiples of 6 found in STEP_9 and calculate the HCF of each pair until we find a pair that has an HCF of 6.
STEP 13
Pairing 90 with 6, we find that the HCF is 6, which meets our condition. So one possible pair is (6, 90).
STEP 14
We need to find another pair. Since we have already used 90, let's try the next highest multiple of 6 which is 54. We need to find another factor of 540 that when paired with 54 gives us an HCF of 6.
STEP 15
Pairing 54 with 10, we find that the HCF is not 6. However, pairing 54 with 20, we find that the HCF is 2, which does not meet our condition.
STEP 16
Continue this process until we find a pair that has an HCF of 6. Pairing 54 with 60, we find that the HCF is 6, which meets our condition. So another possible pair is (54, 60).
STEP 17
We have now found two pairs of numbers that Scott could be thinking of: (6, 90) and (54, 60). Both pairs have an LCM of 90 and an HCF of 6.
The two possible numbers that Scott could be thinking of are 6 and 90, or 54 and 60.
Was this helpful?
