Every multiple of 4 is even because 4 itself is even. Since 4 equals 2 × 2, multiplying anything by 4 automatically builds in a factor of 2, which is the only requirement for a number to be even. This holds for every multiple of 4, no exceptions, whether the number is positive, negative, or zero.
What Makes a Number Even
A number is even if it can be divided by 2 with nothing left over. Mathematically, that means any even number can be written as 2 times some whole number. For example, 10 is even because it equals 2 × 5. The number 36 is even because it equals 2 × 18. If you can express a number in that “2 times something” form, it’s even. That’s the entire test.
Why 4 Guarantees a Factor of 2
The number 4 breaks down into prime factors as 2 × 2. That’s it. There’s no other way to factor it into primes. So when you multiply 4 by any whole number, the result always contains at least one factor of 2 (it actually contains at least two).
Here’s what that looks like with algebra. Take any whole number and call it n. Multiplying it by 4 gives you 4n, which you can rewrite as 2 × (2n). The expression 2n is itself a whole number, so 4n fits the exact definition of an even number: it’s 2 times a whole number. This works regardless of what n is. Pick 1, and you get 4, which is 2 × 2. Pick 7, and you get 28, which is 2 × 14. Pick 1,000,003, and you get 4,000,012, which is 2 × 2,000,006.
The Divisibility Chain
There’s a general principle at work here called transitivity of divisibility. If a smaller number divides evenly into a larger one, then it also divides into every multiple of that larger number. Since 2 divides evenly into 4, it must divide evenly into 4 × 1, 4 × 2, 4 × 3, and so on forever. You can think of it as a chain: 2 goes into 4, so 2 goes into anything that 4 goes into.
This same logic applies beyond 4. Every multiple of 6 is even, every multiple of 10 is even, and every multiple of 248 is even, all for the same reason. Any number that has 2 as a factor will pass that factor of 2 along to all of its multiples.
Why Odd Multipliers Don’t Change Anything
A common source of confusion is multiplying 4 by an odd number. If you multiply 4 × 3, you get 12. If you multiply 4 × 99, you get 396. These results might “feel” like they could be odd since one of the inputs was odd, but the rule for multiplication and parity is straightforward: even times anything is always even. The even factor dominates. Since 4 is even, 4 times any integer, odd or even, produces an even result every time.
Contrast this with a number like 3. Multiples of 3 include 3, 6, 9, 12, 15, and so on, alternating between odd and even. That’s because 3 has no factor of 2 to guarantee evenness. The number 4, with its two factors of 2, never has that problem.
A Quick Way to Spot Multiples of 4
For large numbers, you can check whether something is a multiple of 4 by looking at just the last two digits. If those two digits form a number divisible by 4, the whole number is too. For instance, 1,736 is a multiple of 4 because 36 ÷ 4 = 9. And since you now know every multiple of 4 is guaranteed to be even, you can also confirm the number is even just by checking that last digit: 6, which is even. Both checks will always agree.
This shortcut works because 100 is itself a multiple of 4, so everything above the last two digits is automatically divisible by 4. Only the final two digits determine the remainder.
The Short Version
Four equals 2 × 2. Multiply it by any whole number, and those factors of 2 carry through. Since having at least one factor of 2 is exactly what makes a number even, no multiple of 4 can ever be odd. It’s baked into the structure of the number itself.