To find the two numbers from the given information, we can use the relationship between the highest common factor (HCF), least common multiple (LCM), and the numbers themselves.
We have:
- The highest common factor (HCF) of the two numbers is 23.
- The least common multiple (LCM) includes two factors: 13 and 14.
Firstly, we need to calculate the least common multiple (LCM). The LCM can be obtained by multiplying the HCF and the co-prime factors that help us reach LCM, which can be represented as follows:
LCM = (Number 1 × Number 2) / HCF
Since we know the HCF is 23 and we are looking for the LCM that includes the factors 13 and 14, we can first calculate:
LCM = 13 × 14 = 182
Thus, we can relate this back to our two unknown numbers:
HCF × LCM = Number 1 × Number 2
This gives us:
23 × 182 = Number 1 × Number 2
Now, let’s calculate:
23 × 182 = 4186
This means:
Number 1 × Number 2 = 4186
From the HCF of 23, we know that both numbers must be multiples of 23. Therefore, we can express each number as:
Number 1 = 23a and Number 2 = 23b
Substituting into the equation:
23a × 23b = 4186
This simplifies to:
529ab = 4186
Dividing by 529:
ab = 4186 / 529 = 7.9
Since we want a and b to be integers, we consider possible integer pairs:
The prime factors of 182 are 2, 7, and 13.
Upon checking integer multiples of 23, we find:
- Number 1 = 23 × 2 = 46
- Number 2 = 23 × 7 = 161
Now, let’s verify:
The HCF of 46 and 161 is 23. And as for the LCM:
LCM(46, 161) = (46 × 161) / 23 = 4186 / 23 = 182, which aligns perfectly since we identified 13 and 14 as factors of 182.
Thus, the two numbers satisfying both conditions are:
- 46
- 161
In conclusion, the two numbers are 46 and 161.