# Solution: Finding the Difference Between Two Numbers
When two numbers have a known sum and product, we can use algebraic identities to find their difference without calculating the numbers individually.
Step 1: Set Up the Algebraic Identity
Let the two numbers be \(a\) and \(b\). We know their sum and product, and we can use the identity that relates these to the difference.
We use the identity: \[(a - b)^2 = (a + b)^2 - 4ab\]
Step 2: Calculate the Difference
Substitute the given values into the identity.
The difference between the two numbers is 4.
Answer: (B) 4
Let numbers be 3k and 5k where HCF(3k, 5k) = k = 4. Numbers are 12 and 20. Sum = 32.
For divisibility by 11: alternate sum of digits. 12321: (1+3+1) - (2+2) = 5 - 4 = 1. Recheck: (1+3+1) - (2+2) = 5-4=1. Actually 1-2+3-2+1 = 1. Try: 1-2+3-2+1 = 1. Check: 12321/11 = 1120.09... Correct: (2+2) - (1+3+1) = 4-5 = -1, still divisible.
We need to find a number where the sum of the number and its reciprocal equals 2.5.
Step 1: Set up the equation
Let the number be \(x\). The reciprocal is \(\frac{1}{x}\).
Step 2: Clear the fraction
Multiply both sides by \(x\):
Rearrange to standard quadratic form:
Step 3: Apply the quadratic formula
Using \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = -2.5\), \(c = 1\):
Step 4: Find both solutions
Verification:
- For \(x = 2\): \(2 + \frac{1}{2} = 2.5\) ✓
- For \(x = \frac{1}{2}\): \(\frac{1}{2} + 2 = 2.5\) ✓
Answer: The number is \(2\) or \(\frac{1}{2}\) (Option B)
Let number = x. Then x - x/3 = 30. So 2x/3 = 30, x = 45
Divisors of 16: 1,2,4,8,16. Product = 1×2×4×8×16 = 1024. Or use formula: if n has d divisors, product = n^(d/2). Here d=5, so product = 16^2.5 = 1024
Let numbers be x and x+2. Then x(x+2) = 195. x² + 2x - 195 = 0. Using formula: x = 13. Numbers are 13 and 15. Sum = 28
2^a = 32 = 2^5, so a = 5. And 3^b = 81 = 3^4, so b = 4. Therefore a + b = 5 + 4 = 9
Check 22: 22÷9 = 2 rem 4 ✓, 22÷7 = 3 rem 1 ✗. Check 31: 31÷9 = 3 rem 4 ✓, 31÷7 = 4 rem 3 ✓. Answer is 31 (B, not A). Correction: Option B is correct.
144 = 2⁴ × 3². Odd divisors come only from 3² = (2+1) = 3 odd divisors: 1, 3, 9.
