What is the integer of 285?

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Division of the numbers

If it is: 12/4 = 3, remainder 0 and 15/4 = 3, remainder 3, we say 12 divides by 4 but 15 doesn't divide by 4. So 4 is the divisor of 12 but not from 15.

A natural number "p" shares another number "q" if there is a third natural number "r" with "q = r × p". It is also said that "p" is a factor of "q".

0 is divisible by any number. Any number a, indefinitely from 0, is divisible by 1 and with itself - these are called improper divisors.

Partial rules

The number 84 is divided by 4 and 3, and can also be divided by 4 * 3 = 12. This is not true if the two factors between them are prime. In general, if a is divisible by m and n and gcd (m, n) = 1, then this is also divisible by m × n.

Establishing the divisors, i.e. identifying that one number is divisible by another, is used a great deal to simplify fractions.

The established rules for finding the divisors are based on the case that the numbers are written in the decimal system. The ten multiples divide by 2 and 5 because 10 divides by 2 and 5; multiples of 100 are divisible by 4 and 25, because 100 is divided by 4 and 25; the multiples of 1,000 divide by 8 because 1,000 divides by 8. All powers of 10, when divided by 3 and 9 have the remainder equal to 1.

Because of the rules with remainder operations, when dividing by 3 and 9 we have the following remainder: 600 has a remainder equal to 6 = 1 × 6 (1 for every hundred); 240 = 2 × 100 + 4 × 10, then the remainder is equal to 2 × 1 + 4 × 1 = 6. When dividing a number at 3 or 9 the remainder is equal to that obtained by dividing the total numbers by their number 3 or 9. 7,309 has the sum of the numbers 7 + 3 + 0 + 9 = 19, which divides by both 3 and 9 without a remainder. So 7.309 is not divided by both 3 and 9.

All even powers of 10, 100, 10,000, 1,000,000 etc, when divided by 11 have an equal remainder with 1, and the odd powers of 10, when divided by 11 have an equal remainder with 10 or 10 - 11 = -1 . In this case, the alternate sum of the numbers has the same remainder as the number. How the alternate sum is calculated is shown in the example below.

Example. 85.976: 8 + 9 + 6 = 23 + 5 + 7 = 12, alternating the sum of the numbers. 23 - 12 = 11. So 85.976 is divided by 11.

A number is divided by:
  • 2 if the last number is divisible by 2
  • 4 if the last two numbers form a number divisible by 4;
  • 8 if the last three numbers form a number divisible by 8;
  • 5 if the last number is divisible by 5, i.e. 5 and 0;
  • 25 if the last two numbers form a number divisible by 25;
  • 3 if the sum of the numbers divides by 3;
  • 9 if the sum of the numbers divides by 9;
  • 11 if the alternating sum of the numbers is divided by 11.