small
Lv 7
small asked in Science & MathematicsMathematics · 4 weeks ago

# How to determine if a large number (say in quintillion, 19 digits) is perfectly divisible by 999 without using the process of division?

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• 4 weeks ago

999999 is divisible by 999

999999999 is divisible by 999

In general

10^(3t) - 1 is divisible by 999

So let's say you have a number like 1,378,492,265,446,978,132.  Is it divisible by 999?  Well, we know that 999,999,999,999,999,999 is divisible by 999, so we can start there

1,378,492,265,446,978,132 - 999,999,999,999,999,999 =>

378,492,265,446,978,133

999,999,999,999,999 * 378 = 378,000,000,000,000 - 378

378,492,265,446,978,133 - (378,000,000,000,000 - 378) =>

492,265,446,978,133 + 378 =>

492,265,446,978,511

999,999,999,999 * 492 = 492,000,000,000,000 - 492

265,446,978,511 + 492 = 265,446,979,003

Do you see a pattern emerging?

1 , 378 , 492 , 265 , 446 , 978 , 132

Break it up and add each 3 digit section

1 + 378 + 492 + 265 + 446 + 978 + 132 =>

379 + 757 + 1424 + 132 =>

2692

2 , 692

692 + 2 = 694

694 is not divisible by 999, so 1,378,492,265,446,978,132 is not divisible by 999.

Let's use a number that is divisible by 999 and see if it works:

‭1 , 488 , 038 , 328 , 957 , 537 , 648‬

1 + 488 + 038 + 328 + 957 + 537 + 648 = 2 , 997

2 + 997 = 999

Looks like it works.

• geezer
Lv 7
4 weeks ago

Add up the digits .. if that is divisable by 3 then the number is divisable by 999

• JOHN
Lv 7
4 weeks ago

• ?
Lv 7
4 weeks ago

If the sum of the digits are divisible by 3, then the large number is divisible by 999.