Find the H.C.F and L.C.M of 480 and 720 using the Prime factorisation method. The H.C.F of 85 and 238 is expressible in the form 85m - 238. Find the value of m.

Steps to Solve

1. Prime Factorization of 480 To find the prime factors of 480, we divide by the smallest prime numbers until we reach 1:

   • $480 \div 2 = 240$
   • $240 \div 2 = 120$
   • $120 \div 2 = 60$
   • $60 \div 2 = 30$
   • $30 \div 2 = 15$
   • $15 \div 3 = 5$
   • $5 \div 5 = 1$

   Therefore, the prime factorization of 480 is: $$ 480 = 2^5 \times 3^1 \times 5^1 $$

2. Prime Factorization of 720 Similarly, we find the prime factors of 720:

   • $720 \div 2 = 360$
   • $360 \div 2 = 180$
   • $180 \div 2 = 90$
   • $90 \div 2 = 45$
   • $45 \div 3 = 15$
   • $15 \div 3 = 5$
   • $5 \div 5 = 1$

   Therefore, the prime factorization of 720 is: $$ 720 = 2^4 \times 3^2 \times 5^1 $$

3. Finding the H.C.F. The Highest Common Factor (H.C.F) is found by taking the lowest powers of the common prime factors:

   • For $2$: $\min(5, 4) = 4$
   • For $3$: $\min(1, 2) = 1$
   • For $5$: $\min(1, 1) = 1$

   Thus, the H.C.F is: $$ \text{H.C.F} = 2^4 \times 3^1 \times 5^1 = 48 $$

4. Finding the L.C.M. The Least Common Multiple (L.C.M) is found by taking the highest powers of all prime factors:

   • For $2$: $\max(5, 4) = 5$
   • For $3$: $\max(1, 2) = 2$
   • For $5$: $\max(1, 1) = 1$

   Thus, the L.C.M is: $$ \text{L.C.M} = 2^5 \times 3^2 \times 5^1 = 720 $$