What you have here is math from another world! O_o
It called a change of base, we use base ten (digits 0 - 9) because we have 10 digits
the first step is figuring out what base these numbers are in.
usually 435 for example equals...
4 x 10^2 = 400 +
3 x 10^1 = 030 +
5 x 10^0 = 005 =
hence 435...
At a glance this base system must have over 7 digits (because 0 - 6 is only 7 digits and these numbers have a 7 in them).
So now we guess!
In base 8, 137 should equal...
X x 8^2 + Y x 8^1 + Z x 8^0
but we want to convert these numbers back to base 10 to see if they really work... So
1 x 8^2 = 64 +
3 x 8^1 = 24 +
7 x 8^0 = 7 = 95
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2 x 8^2 = 128 +
7 x 8^1 = 56 +
6 x 8^0 = 6 = 190 (I converted both 137 and 276 to base 10, I'll explain why in a moment)
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4 x 8^2 = 256 +
3 x 8^1 = 24 +
5 x 8^0 = 5 = 285
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These numbers are the same as 137, 276 and 435 in the base 10. Very important to remember that because if they are the same numbers they should be able to do this...
137 + 276 = 435 (substitute the numbers)
95 + 190 = 285 and it checks out beautifully! Lucky guess that the system was base 8 because it will not work like this is you replaced the base number with some number like 9 for example (base number being the number with the exponent [X x Y^z ; Y is base]
So now you must convert the number in the next equation to base 10
7 x 8^2 = 448 +
3 X 8^1 = 24 +
1 X 8^0 = 1 = 473
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6 x 8^2 = 384 +
7 x 8^1 = 56 +
2 x 8^0 = 2 = 442
Now you have both number in base 10 so add them and get 915.
This last part is kind of tricky but to change the number back to base 8 (which is what the answer is requiring) you must find the largest exponent of 8 that goes into the number 915. 8^3 is that exponent. It goes in once so lets place a one in your new number. 1, _ _ _ (the number will be 4 digits great because between powers 3 and 0 there are 4 digits).
Keep that there is a remainder of 403. Don't get rid of it! Now take the next greatest exponent of 8 and divide the remainder by that number (aka 403 / 64). It goes in 6 times. That's your next digit. 1,6 _ _. The remainder is 19. The next greatest exponent of 8 is 1 so 19 / 8 which gives you 2 with a remainder of 3. So thus far is 1,62 _. Lastly 8^0 (which equal one) is the last exponent. So that last remainder is your last digit. Here's the big moment. THE FINAL NUMBER IS 1,623! You will know if you changed base properly if there is no digit greater than the base you are changing to, meaning if we had an answer with the digit 9 in it, we would be in the wrong base.
Great problem. Rough explanation. Changing bases is an advanced arithmetic property and procedure unfortunately not taught much in schools anymore. But it proves crazy things like 137+276 can equal 435!
