Have your friends pick any, say, four-digit number. (These rules apply to any whole numbers, but the trick becomes more difficult the more digits there are.)
Announce that you can not only tell, on sight, whether the chosen number divides by 9, but you can also tell, on sight, what the remainder is if it does not divide by 9.
Your pals pick, say, 1548. You do not test-divide. Instead you add the digits:
1+5+4+8 = 18. You know 18 divides by 9. But you can take the next step: 1+8 = 9.
Assuming you can add simple sums fairly quickly in your head, you can now announce that 1548 is divisible by 9. Have your buddies check by hand division or by calculator. [There are mathematical proofs that any whole number (integer) with digits that sum, by repeated summing, to 9 are divisible by 9. We won't go into that here.]
Now suppose they next try the number 7237. You sum up 7+2+3+7 = 19. Then 1+9 = 10. Then 1+0 = 1. You announce, without taking very long and without any scribbles or calculator work, that if 7237 is divided by 9, the remainder will be 1.
This is probably faster to check by hand division than by routine calculator.
Old school hand method
Rudimentary calculator method
You can even challenge your maties to try a random number, as generated on a calculator or on the internet or by some other means (such as taking the first four digits from a book's ISBN or from some serial number on a food package).
Announce that you can not only tell, on sight, whether the chosen number divides by 9, but you can also tell, on sight, what the remainder is if it does not divide by 9.
Your pals pick, say, 1548. You do not test-divide. Instead you add the digits:
1+5+4+8 = 18. You know 18 divides by 9. But you can take the next step: 1+8 = 9.
Assuming you can add simple sums fairly quickly in your head, you can now announce that 1548 is divisible by 9. Have your buddies check by hand division or by calculator. [There are mathematical proofs that any whole number (integer) with digits that sum, by repeated summing, to 9 are divisible by 9. We won't go into that here.]
Now suppose they next try the number 7237. You sum up 7+2+3+7 = 19. Then 1+9 = 10. Then 1+0 = 1. You announce, without taking very long and without any scribbles or calculator work, that if 7237 is divided by 9, the remainder will be 1.
This is probably faster to check by hand division than by routine calculator.
Old school hand method
9|7237
9|72 = 8
72-72 = 0
9|3 = 0
9|37 = 4
37-36 = 1
So 9|7237 = 804 r 1.
Rudimentary calculator method
7237/9 = 804.111... Technically 0.111... to infinity is another name for 1, so you can stop now if you know that. But not all digit extensions are so simple. In that case, we continue with
9x804 = 7236. Hence 7237 - 7236 = 1, and the remainder is verified.
You can even challenge your maties to try a random number, as generated on a calculator or on the internet or by some other means (such as taking the first four digits from a book's ISBN or from some serial number on a food package).
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