Even gap, same parity

The idea

If the difference between two numbers is even, they have the same parity: either both are even or both are odd.

Two numbers, a gap of 6

Two numbers are 6 apart. One of them is 9.

What can you say about the other number?

• It is odd
• It is even
• We cannot tell

Walk from one number to another

Tap a number, then another one. The toy walks from the first to the second. Watch the colour at every step.

An even walk comes back to the same colour

If the difference between two numbers is even, they have the same parity: either both are even or both are odd.

Every step along the numbers swaps the colour: blue, orange, blue, orange … An even number of steps swaps an even number of times, so we land on the same colour.

It also works the other way round: if one number is even and the other odd, their difference must be odd.

Notes

If the difference between two numbers is even, they have the same parity: either both are even or both are odd.

Check yourself

Two numbers differ by 18. One of them is 41. What is the other number?

Look at 3, 8, 12, 15 and 20. How many pairs of these numbers have an even difference?

Answer: 4

Even numbers: 8, 12, 20 make 3 pairs. Odd numbers: 3, 15 make 1 pair. Pairs of the same parity: 3 + 1 = 4.

Without subtracting: is 2891 − 1473 even or odd?

The difference of two numbers is 35. What must be true?

• Even. Look at the gap: 18 is even, so we land on the same colour as 41. Both must be odd.
• Odd — correct. Yes! The gap 18 is even, so the other number has the same parity as 41: it is 59 or 23, both odd.
• We cannot tell. We can tell. An even gap keeps the parity, so the other number is odd like 41.
• Even — correct. Yes! Both numbers are odd, so the difference is even. (2891 − 1473 = 1418.)
• Odd. 2891 ends in 1 and 1473 ends in 3. Both numbers are odd, so their difference is even.
• It depends on the digits in the middle. Only the last digit decides if a number is odd or even, and both numbers end in an odd digit.
• Both are even. The gap 35 is odd. An odd gap swaps the colour, so the two numbers cannot have the same parity.
• Both are odd. The gap 35 is odd. An odd gap swaps the colour, so the two numbers cannot have the same parity.
• One is even and the other is odd — correct. Yes! If they had the same parity, the difference would be even. 35 is odd, so the parities differ.