Why Numbers Divisible by 10 Are Always Even: A Mathematical Proof

Let's prove that every integer divisible by 10 must be an even number. To demonstrate this, we'll show that if an integer is divisible by 10, then it's also divisible by 2, the definition of an even number.

Proof:

1. Assumption: Let's assume there exists an integer 'n' that is divisible by 10. This means there exists another integer 'k' such that n = 10k.

2. Factorization: Since n = 10k, we can rewrite it as n = (2 * 5)k.

3. Case Analysis: Now, let's consider the possible values of 'k':

   • Case 1: 'k' is even. If 'k' is an even integer, we can express it as k = 2m, where 'm' is another integer. Substituting this into n = (2 * 5)k, we get n = (2 * 5)(2m) = 4(5m). This shows that 'n' is divisible by 2, making it an even number.

   • Case 2: 'k' is odd. If 'k' is an odd integer, we can express it as k = 2m + 1, where 'm' is an integer. Substituting this into n = (2 * 5)k, we get n = (2 * 5)(2m + 1) = 4m + 2 = 2(2m +1). Again, 'n' is divisible by 2, confirming it's an even number.

Conclusion:

In both cases, we've shown that if 'n' is divisible by 10, it must be an even number. This holds true whether 'k' (the other factor of 'n') is even or odd.

Therefore, we have proven that every integer divisible by 10 must be an even number.