What happens to the HCF of two numbers if both numbers are doubled? Take some pairs of numbers and explore. Are you able to see why the HCF will also double?

Question

Homi · Accepted Answer

The HCF of two numbers also doubles when both numbers are doubled.

Step 1 — Exploring with examples

Let us take two numbers. We will find their HCF. Then we double both numbers. We find the HCF of the new numbers. We compare the two HCFs.

Let us take the numbers 6 and 9. We find the HCF of 6 and 9. We write the prime factors of 6. $6 = 2 \times 3$ We write the prime factors of 9. $9 = 3 \times 3$ The common prime factor is 3. So, the HCF of 6 and 9 is 3.

Now, we double both numbers. The first new number is $2 \times 6$ . $2 \times 6 = 12$ The second new number is $2 \times 9$ . $2 \times 9 = 18$ We find the HCF of 12 and 18. We write the prime factors of 12. $12 = 2 \times 2 \times 3$ We write the prime factors of 18. $18 = 2 \times 3 \times 3$ The common prime factors are 2 and 3. We multiply these common prime factors. The HCF of 12 and 18 is $2 \times 3$ . $2 \times 3 = 6$ The new HCF is 6. We compare the original HCF (3) and the new HCF (6). The new HCF is double the original HCF.

Let us take another pair of numbers. We will use 10 and 15. We find the HCF of 10 and 15. We write the prime factors of 10. $10 = 2 \times 5$ We write the prime factors of 15. $15 = 3 \times 5$ The common prime factor is 5. So, the HCF of 10 and 15 is 5.

Now, we double both numbers. The first new number is $2 \times 10$ . $2 \times 10 = 20$ The second new number is $2 \times 15$ . $2 \times 15 = 30$ We find the HCF of 20 and 30. We write the prime factors of 20. $20 = 2 \times 2 \times 5$ We write the prime factors of 30. $30 = 2 \times 3 \times 5$ The common prime factors are 2 and 5. We multiply these common prime factors. The HCF of 20 and 30 is $2 \times 5$ . $2 \times 5 = 10$ The new HCF is 10. We compare the original HCF (5) and the new HCF (10). The new HCF is double the original HCF.

Diagram 1

Step 2 — Why the HCF doubles

Let us understand why this happens. Let the two original numbers be $a$ and $b$ . Let their Highest Common Factor be $H$ . This means $H$ is the largest number that divides both $a$ and $b$ . We can write $a$ as $H \times x$ . We can write $b$ as $H \times y$ . Here, $x$ and $y$ are numbers that have no common factors other than 1. This is important for $H$ to be the highest common factor.

Now, we double both numbers. The new first number becomes $2 \times a$ . $2 \times a = 2 \times (H \times x)$ $= (2 \times H) \times x$ The new second number becomes $2 \times b$ . $2 \times b = 2 \times (H \times y)$ $= (2 \times H) \times y$ Look at the new numbers: $(2 \times H) \times x$ and $(2 \times H) \times y$ . We can see that $2 \times H$ is a common factor for both new numbers. Since $x$ and $y$ have no common factors (except 1), then $2 \times H$ must be the highest common factor. So, the HCF of the doubled numbers is $2 \times H$ . This means the HCF has also doubled.

Answer

(i) When both numbers are doubled, their HCF also doubles. (ii) For example, HCF(6, 9) is 3. When doubled to 12 and 18, HCF(12, 18) is 6. (iii) The original HCF ( $H$ ) becomes $2H$ because $2H$ is the largest common factor of $2a$ and $2b$ .

Question 17

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