One even, one odd

The idea

The sum of an even number and an odd number cannot be arranged in pairs and so is odd; hence the sum of any two consecutive numbers is odd.

Maria’s big sum

Martin and Maria were born exactly one year apart. Today Maria says: “The sum of our ages is 112!”

Is that possible?

• Yes, it is possible
• No, it is not possible
• It depends on their birthdays

Even plus odd

First pick one even and one odd number and save the sum. Then try pairs of consecutive numbers like Martin and Maria.

The leftover stays

The even number is all pairs. The odd number has one dot left over. Putting them together, that one dot still has no partner. So the total is odd.

Two consecutive numbers are one even and one odd, so their sum is always odd. Maria’s 112 is even. It is impossible.

The sum of an even number and an odd number cannot be arranged in pairs, so it is odd. Hence the sum of any two consecutive numbers is odd.

• Even + odd | Total
• 6 + 7 | 13, odd
• 12 + 5 | 17, odd
• 51 + 52 (consecutive) | 103, odd

Notes

The sum of an even number and an odd number cannot be arranged in pairs, so it is odd. Hence the sum of any two consecutive numbers is odd.

Check yourself

Which total can be got by adding an even number and an odd number?

Two sisters were born exactly one year apart. Could their ages add up to 90?

Two ages one year apart add up to 77. What is the older age?

Answer: 39

38 + 39 = 77. The older is 39.

Ravi adds an even number and an odd number and says the answer is 48. What do you say?

• 35 — correct. Yes! For example 20 + 15 = 35. One dot is left over, so the total is odd.
• 36. 36 is even. An even number and an odd number never make an even total.
• 48. 48 is even. The odd number’s single dot is always left over.
• 100. 100 is even. Even plus odd is always odd.
• Yes: 45 and 45. The sisters are one year apart, so their ages cannot both be 45.
• No: consecutive ages add up to an odd number — correct. Yes! Their ages are one even and one odd, so the sum is odd, not 90.
• Yes: 44 and 46. Those ages are two years apart. One year apart means 44 and 45, or 45 and 46.
• Fine: 48 is even because one of the numbers is even. One even number is not enough: the odd number’s single dot is still left over.
• He must have added two odd numbers. Two odd numbers do give an even total, but we know he added an even and an odd number. His sum cannot be 48.
• He made a mistake: even + odd is always odd — correct. Yes! 48 is even, and even + odd can never be even.