A number is three times another number and their HCF is 8. What is the sum of the squares of the numbers?
640
Finding Numbers Using HCF and Ratio
This problem asks us to find two numbers given a relationship between them and their Highest Common Factor (HCF). We are told that one number is three times another number, and their HCF is 8. We then need to calculate the sum of the squares of these two numbers.
Understanding the Relationship and HCF
Let's represent the two unknown numbers. We are given that one number is three times another. Let the smaller number be represented by a variable, say $a$. Then the larger number will be $3a$.
We are also given that the HCF of these two numbers is 8.
Using the Property of HCF
The HCF of two numbers is the largest positive integer that divides both numbers without leaving a remainder. When one number is a multiple of another, the HCF of the two numbers is the smaller number itself.
In our case, the numbers are $a$ and $3a$. Since $3a$ is a multiple of $a$, the HCF of $a$ and $3a$ is $a$.
We are given that the HCF is 8. Therefore, we can set the HCF we found equal to the given HCF:
Determining the Numbers
Now that we know the value of $a$, we can find the two numbers:
- The smaller number is $a = 8$.
- The larger number is $3a = 3 \times 8 = 24$.
So, the two numbers are 8 and 24.
Let's quickly check their HCF. The factors of 8 are 1, 2, 4, 8. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The common factors are 1, 2, 4, 8. The highest common factor is indeed 8.
Calculating the Sum of Squares
The problem asks for the sum of the squares of these numbers. We need to square each number and then add the results.
- Square of the smaller number ($8$): $$8^2 = 8 \times 8 = 64$$
- Square of the larger number ($24$): $$24^2 = 24 \times 24 = 576$$
Now, add the squares together:
Performing the addition:
The sum of the squares of the numbers is 640.
Final Answer
The sum of the squares of the two numbers is 640.
| Description | Value |
|---|---|
| Smaller Number (a) | 8 |
| Larger Number (3a) | 24 |
| Square of Smaller Number ($8^2$) | 64 |
| Square of Larger Number ($24^2$) | 576 |
| Sum of Squares ($64 + 576$) | 640 |
Revision Table: Key Concepts
| Concept | Explanation | Application in Problem |
|---|---|---|
| Definition of HCF | The largest number that divides two or more numbers exactly. | Given HCF is 8. |
| HCF of a Number and its Multiple | HCF($n$, $kn$) = $n$, where $k$ is an integer > 1. | HCF($a$, $3a$) = $a$. This allowed us to find $a=8$. |
| Squaring a Number | Multiplying a number by itself ($n^2 = n \times n$). | Calculated $8^2$ and $24^2$. |
| Sum of Squares | Adding the results of squaring two or more numbers. | Calculated $8^2 + 24^2$. |
Additional Information: HCF and LCM
While this problem focused on HCF, it's useful to understand its relationship with the Least Common Multiple (LCM).
- HCF: The greatest common divisor. Used for sharing things equally or arranging items in equal rows/groups.
- LCM: The smallest positive integer that is a multiple of two or more numbers. Used for events that repeat at regular intervals (like buses arriving or lights flashing) to find when they happen together again.
For any two positive integers, say $x$ and $y$, there is a fundamental relationship between their HCF and LCM:
In our problem, the numbers are 8 and 24. Their HCF is 8. We can find their LCM using this property:
The LCM of 8 and 24 is 24. This makes sense because 24 is a multiple of 8, so the smallest number that is a multiple of both 8 and 24 is 24.
Important Questions from LCM and HCF
The greatest three-digit number which is divisible by 14, 28, and 42 is:
What is the greatest number that will divide 209 and 347 leaving remainder 5 and 7 respectively?
The HCF of 2091, 3485 and 4879 is x. The sum of the digits of x is:
If three numbers are in ratio of 3 : 5 : 7 and their LCM is 2415, what is the difference between the second number and the first number?
The least number of square tiles required to pave the floor of a room with dimensions 15 meters and 12 meters will be:
