Let's solve this problem step by step.
We know that the highest common factor (HCF) of the two numbers is 2, and the least common multiple (LCM) is 70.
To find the two numbers, we can start by listing the multiples of 70 and identifying the ones that fall within the given range (less than 15 and greater than 7):
Multiples of 70: 70, 140, 210, 280, 350, 420, 490, 560, 630, 700, 770, 840, 910, ...
Since the numbers must be less than 15, we can eliminate all the multiples of 70 beyond 14. Now let's consider the multiples of 70 that are greater than 7:
Multiples of 70 between 7 and 14: 70, 140
Now, let's consider the common factors of these two multiples of 70. The common factors of two numbers are the numbers that divide both of them without leaving a remainder.
Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70
The highest common factor of the two numbers is 2. Since the highest common factor is 2, we can conclude that the other factors (besides 2) cannot be common factors of the two numbers.
Therefore, the two numbers that Matthew is thinking of are 70 and 140.