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"What is the remainder when 582 is divided by 3?"

I want to know how can I solve questions like the one displayed in the image above whether they have a remainder or not, quickly and without the use of a calculator, with paper drafts and without them.

I am not very sharp in mathematics so I would be very grateful if someone could elaborate with me on issues like these.

MachineLearner
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Michael Vainshtein
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    There's a famous criterion to determine whether a number is divisible by $3$. Are you aware of it? – Kman3 Oct 01 '21 at 16:02
  • Actually this task is so easy that you can directly solve check it. Note that $582 = 300 + 270 + 12$. What is the remainder of each term? – MachineLearner Oct 01 '21 at 16:04
  • @Kman3 No I am not familiar with this criterion, could you please share it with me? – Michael Vainshtein Oct 01 '21 at 16:23
  • Questions "like the one displayed" depend a lot on the divisor. For the divisor $3$ and $9$ the remainder is easily calculated, use the sum of the digits instead. So the remainder in "$582$ divided by $3$" is the same one as in "$5+8+2$ divided by $3$" - and the latter my be simpler... For other "simple divisors" like $2$, $4$, $5$, $8$, $11$ there are more or less easy rules to get the remainder... For $7$ or for $13$ things are not so simple to visualize... – dan_fulea Oct 01 '21 at 16:24
  • @MachineLearner Are you a wizard? that's pretty clever. – Michael Vainshtein Oct 01 '21 at 16:29
  • @MichaelVainshtein As mentioned by others, if the sum of a number's digits is a multiple of $3$, then the number is divisible by $3$. So I can tell immediately that $582/3$ has remainder zero because $5+8+2=15=5(3)$. One straightforward way you could do these problems is with long division, but if you don't have pen and paper I guess you would have to do long division in your head. – Kman3 Oct 01 '21 at 19:31

1 Answers1

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To check divisibility by 3, add the digits of your number. In this case do $5+8+2$ and see if the sum is divisible by 3. If it is, remainder is 0.

For other such rules, see this Wikipedia page: Here

Doobius
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