Question 4

We find the HCF by listing the prime factors of each number.

Step 1 — HCF of 24 and 180

Let us find the prime factors of 24. $24 = 2 \times 12$ $= 2 \times 2 \times 6$ $= 2 \times 2 \times 2 \times 3$ So, the prime factors of 24 are $\mathbf{2, 2, 2, 3}$.

Next, we find the prime factors of 180. $180 = 2 \times 90$ $= 2 \times 2 \times 45$ $= 2 \times 2 \times 3 \times 15$ $= 2 \times 2 \times 3 \times 3 \times 5$ So, the prime factors of 180 are $\mathbf{2, 2, 3, 3, 5}$.

Now, we find the common prime factors. The common prime factors are 2, 2, and 3. We multiply these common factors together. $HCF(24, 180) = 2 \times 2 \times 3$

$\boxed{\textbf{12}}$

Step 2 — HCF of 42, 75, and 24

Let us find the prime factors of 42. $42 = 2 \times 21$ $= 2 \times 3 \times 7$ So, the prime factors of 42 are $\mathbf{2, 3, 7}$.

Next, we find the prime factors of 75. $75 = 3 \times 25$ $= 3 \times 5 \times 5$ So, the prime factors of 75 are $\mathbf{3, 5, 5}$.

Then, we find the prime factors of 24. $24 = 2 \times 12$ $= 2 \times 2 \times 6$ $= 2 \times 2 \times 2 \times 3$ So, the prime factors of 24 are $\mathbf{2, 2, 2, 3}$.

Now, we find the common prime factors for all three numbers. The only common prime factor is 3. $HCF(42, 75, 24) = 3$

$\boxed{\textbf{3}}$

Step 3 — HCF of 240 and 378

Let us find the prime factors of 240. $240 = 2 \times 120$ $= 2 \times 2 \times 60$ $= 2 \times 2 \times 2 \times 30$ $= 2 \times 2 \times 2 \times 2 \times 15$ $= 2 \times 2 \times 2 \times 2 \times 3 \times 5$ So, the prime factors of 240 are $\mathbf{2, 2, 2, 2, 3, 5}$.

Next, we find the prime factors of 378. $378 = 2 \times 189$ $= 2 \times 3 \times 63$ $= 2 \times 3 \times 3 \times 21$ $= 2 \times 3 \times 3 \times 3 \times 7$ So, the prime factors of 378 are $\mathbf{2, 3, 3, 3, 7}$.

Now, we find the common prime factors. The common prime factors are 2 and 3. We multiply these common factors together. $HCF(240, 378) = 2 \times 3$

$\boxed{\textbf{6}}$

Step 4 — HCF of 400 and 2500

Let us find the prime factors of 400. $400 = 2 \times 200$ $= 2 \times 2 \times 100$ $= 2 \times 2 \times 2 \times 50$ $= 2 \times 2 \times 2 \times 2 \times 25$ $= 2 \times 2 \times 2 \times 2 \times 5 \times 5$ So, the prime factors of 400 are $\mathbf{2, 2, 2, 2, 5, 5}$.

Next, we find the prime factors of 2500. $2500 = 2 \times 1250$ $= 2 \times 2 \times 625$ $= 2 \times 2 \times 5 \times 125$ $= 2 \times 2 \times 5 \times 5 \times 25$ $= 2 \times 2 \times 5 \times 5 \times 5 \times 5$ So, the prime factors of 2500 are $\mathbf{2, 2, 5, 5, 5, 5}$.

Now, we find the common prime factors. The common prime factors are 2, 2, 5, and 5. We multiply these common factors together. $HCF(400, 2500) = 2 \times 2 \times 5 \times 5$ $= 4 \times 25$

$\boxed{\textbf{100}}$

Step 5 — HCF of 300 and 800

Let us find the prime factors of 300. $300 = 2 \times 150$ $= 2 \times 2 \times 75$ $= 2 \times 2 \times 3 \times 25$ $= 2 \times 2 \times 3 \times 5 \times 5$ So, the prime factors of 300 are $\mathbf{2, 2, 3, 5, 5}$.

Next, we find the prime factors of 800. $800 = 2 \times 400$ $= 2 \times 2 \times 200$ $= 2 \times 2 \times 2 \times 100$ $= 2 \times 2 \times 2 \times 2 \times 50$ $= 2 \times 2 \times 2 \times 2 \times 2 \times 25$ $= 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5$ So, the prime factors of 800 are $\mathbf{2, 2, 2, 2, 2, 5, 5}$.

Now, we find the common prime factors. The common prime factors are 2, 2, 5, and 5. We multiply these common factors together. $HCF(300, 800) = 2 \times 2 \times 5 \times 5$ $= 4 \times 25$

$\boxed{\textbf{100}}$

Answer

(a) The HCF of 24 and 180 is 12. (b) The HCF of 42, 75, and 24 is 3. (c) The HCF of 240 and 378 is 6. (d) The HCF of 400 and 2500 is 100. (e) The HCF of 300 and 800 is 100.