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Two Numbers Under the Microscope

Stage: 1 Challenge Level: Challenge Level:2 Challenge Level:2

More than $30$ solutions were sent in but the activity had asked for some kind of proof and many solutions  unfortunately consisted of examples of numbers only, so here are the ones that said a bit more.

Firstly Amrit from Newton Farm Nursery, Infant and Junior School in the United Kingdom who wrote;

Every odd number can be written in the form $2a + 1$, so let the two odd numbers be $2a + 1$ and $2b + 1$. Thus their sum is $2(a + b + 1)$. As every even number can be written in the form $2n$, the sum of two odd numbers is always even.


From James from St. Johns School in Northwood, England we had;

They always add up to an even number because if you add together the even numbers that are one less than each number you'll have an even number. Then if you add together the two that are left you get an even number. If you do this with three odd numbers it would give you an odd number because you'll have three left to add together. If you add up an even number of odd numbers you get an even number but if you have an odd number of odd numbers you get an odd number.


From Madison at Norwayne School in the USA we had:

Any odd number always ends up in pairs of two when added together. Like if you take $3$ and $9$ they end up in pairs of two. Or if you take $5$ and $9$ they end up in pairs of two. The reason for that is if you add any odd number with a other odd number then the sum is even. So you can split the sum up into to even group.


From Peter at the British International School in Istanbul in Turkey we were sent:

An odd number actually is an "even number plus $1$" (or minus $1$). So if you add two odd numbers. Both "$1$"s from both numbers togeher make a "$2$", which is an even number; and the remaining part of the both numbers were even anyway, so the total is always even. This is best shown by the picture of Abdul with the orange and green coloured squares.


Christopher from Lyngby/Lundtofte
 in Denmark sent in some good material but unfortunately I could not open open the file. Thanks to all of you who submitted solutions.