What if you use logical reasoning?

Number Theory Level 3

I have a box which contains x x amount of balls.

  • If I take them out 1 at a time, no balls would be left in the box.

  • If I take them out 2 at a time, 1 ball would be left in the box.

  • If I take them out 3 at a time, no balls would be left in the box.

  • If I take them out 4 at a time, 1 ball would be left in the box.

  • If I take them out 5 at a time, 4 balls would be left in the box.

  • If I take them out 6 at a time, 3 balls would be left in the box.

  • If I take them out 7 at a time, no balls would be left in the box.

  • If I take them out 8 at a time, 1 ball would be left in the box.

  • If I take them out 9 at a time, no balls would be left in the box.

What is the least number of balls I have?


The answer is 1449.

Bloons Qoth by Bloons Qoth , 21, USA

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2 solutions

Saya Suka
Feb 3, 2017

First, let's look at the no-remainder divisions, and those are 1, 3, 7 and 9. So the number we are looking for is a multiple of LCM of the 4 numbers, which is 63. Now look at division by 2 (or any other even number at that), clearly we can see that our number is an odd one. Division by 5 told us that the number would have either 4 or 9 at the ones place, but as we already confirmed before, we are looking for an odd number, so 9 it is. Now if some number with a last digit 9 is a multiple of 63, then the multiplicator (or multiplier or multiplicand) can only have 3 as its own last digit. Thus, 63 x (10n + 3) would produce the desired answer. n = 2 will get us the smallest of answers which is 1449.

1 Helpful 0 Interesting 0 Brilliant 0 Confused 
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for x in range(1000,2000):
    if (x%4==1):
        if (x%5==4):
            if (x%6==3):
                if (x%7==0):
                    if (x%8==1):
                        if (x%9==0):
                            print(x)


1
 
1449

(No output till 1000)

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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