Solved Examples

Example 1: Parity Logic

Question: If $n$ is an integer, prove that $n^2 + n$ is always even.

Solution: We can factor the expression: $n^2 + n = n(n+1)$ This is the product of two consecutive integers. In any pair of consecutive integers, one must be even and the other odd. Since $\text{Even} \times \text{Odd} = \text{Even}$, the product is always even.

Example 2: Finding Divisors

Question: Check if 12345 is divisible by 15.

Solution: For divisibility by 15, the number must be divisible by both 3 and 5.

1. Check 5: The number ends in 5. Yes.
2. Check 3: Sum of digits $= 1 + 2 + 3 + 4 + 5 = 15$. Since 15 is divisible by 3, the number is divisible by 3. Yes. Conclusion: 12345 is divisible by 15.

Example 3: Cryptarithm Logic

Question: Find $A$ and $B$ if $2A \times A = 1B A$.

Solution:

1. Units digit: $A \times A$ ends in $A$. Possible values for $A$: $0, 1, 5, 6$.
2. Test values:
   • $A=0$: $20 \times 0 = 0$ (Not 3 digits).
   • $A=1$: $21 \times 1 = 21$ (Not 3 digits).
   • $A=5$: $25 \times 5 = 125$. Here $B=2$. Fits $1BA$.
   • $A=6$: $26 \times 6 = 156$. Here $B=5$. Fits $1BA$. Answer: Two solutions: $25 \times 5 = 125$ or $26 \times 6 = 156$.