Let a and b be two positive integers such that a = p3q4 and b = p2q3 , where p and q areprime numbers. If HCF(a,b) = pmqn and LCM(a,b) = prqs, then (m+n)(r+s

Sure, let's solve the problem step by step.

Step 1: Expressing a and b in terms of their prime factors: We are given that a = p^3 * q^4 and b = p^2 * q^3, where p and q are prime numbers.

Step 2: Finding the highest common factor (HCF) of a and b: To find the HCF, we need to determine the highest power of each prime factor that appears in both a and b. In this case, the HCF(a, b) = p^m * q^n.

Comparing the powers of p in a and b, we see that the highest power of p that appears in both a and b is p^2. Therefore, m = 2.

Similarly, comparing the powers of q in a and b, we see that the highest power of q that appears in both a and b is q^3. Therefore, n = 3.

So, HCF(a, b) = p^2 * q^3.

Step 3: Finding the least common multiple (LCM) of a and b: To find the LCM, we need to determine the highest power of each prime factor that appears in either a or b. In this case, the LCM(a, b) = p^r * q^s.

Comparing the powers of p in a and b, we see that the highest power of p that appears in either a or b is p^3. Therefore, r = 3.

Similarly, comparing the powers of q in a and b, we see that the highest power of q that appears in either a or b is q^4. Therefore, s = 4.

So, LCM(a, b) = p^3 * q^4.

Step 4: Finding the expression for (m + n)(r + s): Substituting the values of m, n, r, and s into the expression (m + n)(r + s), we get:

(m + n)(r + s) = (2 + 3)(3 + 4) = 5 * 7 = 35.

Therefore, (m + n)(r + s) = 35.