a) 2 numbers have a HCF (highest common factor) of 18 and LCM (lowest common multiple) of 360. What could the 2 numbers be?
b) Emily thinks of 2 numbers. The HCF of the two numbers is 1. The LCM of the 2 numbers is a multiple of 40. Write down 2 possible numbers that Emily could be thinking of.
c) Niamh thinks of 2 numbers. The HCF of the two numbers is 8. The LCM of the two numbers is a multiple of 5. Write down 2 possible numbers that she could be thinking of.
Please help, and include an explanation as I would like to understand it!
a) To find two numbers with a highest common factor (HCF) of 18 and a lowest common multiple (LCM) of 360, we can use the relationship between the HCF and LCM. The product of the HCF and LCM of two numbers is equal to the product of the two numbers themselves.
So, let's assume the two numbers are a and b. We know that HCF(a, b) = 18 and LCM(a, b) = 360. Therefore, we have:
a × b = HCF(a, b) × LCM(a, b) = 18 × 360 = 6480.
Now, we need to find two numbers whose product is 6480. We can start by listing the factors of 6480:
1 × 6480
2 × 3240
3 × 2160
4 × 1620
5 × 1296
6 × 1080
8 × 810
9 × 720
10 × 648
...
We can continue this process until we find two numbers whose product is 6480. In this case, we find the numbers 90 and 72:
90 × 72 = 6480.
Therefore, two possible numbers with a HCF of 18 and a LCM of 360 could be 90 and 72.
b) Emily thinks of two numbers with a HCF of 1 and a LCM that is a multiple of 40. To find two possible numbers, we need to find two numbers whose product is a multiple of 40.
We can start by listing numbers whose product is a multiple of 40:
1 × 40 = 40
2 × 20 = 40
4 × 10 = 40
5 × 8 = 40
Therefore, two possible numbers that Emily could be thinking of are 5 and 8.
c) Niamh thinks of two numbers with a HCF of 8 and a LCM that is a multiple of 5. To find two possible numbers, we need to find two numbers whose product is a multiple of 5 and has 8 as the highest common factor.
We can start by listing numbers whose product is a multiple of 5:
1 × 5 = 5
2 × 5 = 10
3 × 5 = 15
4 × 5 = 20
5 × 5 = 25
...
To have a HCF of 8, we need one of the numbers to be a multiple of 8. Let's try multiplying different numbers by 8 until we find a number that produces a multiple of 5:
8 × 1 = 8 (does not give a multiple of 5)
8 × 2 = 16 (does not give a multiple of 5)
8 × 3 = 24 (does not give a multiple of 5)
8 × 4 = 32 (does not give a multiple of 5)
8 × 5 = 40 (gives a multiple of 5)
Therefore, one possible number that Niamh could be thinking of is 8, and the other number could be 40.
Note that there could be additional pairs of numbers that satisfy the given conditions, but these are two possible examples.