Nine remainder

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The nine remainder of an integer is the remainder it leaves when dividing by 9, i.e. one of the nine natural numbers 0, 1, 2, 3, 4, 5, 6, 7, or 8.

It is the modulus function which determines the remainder of an integer division, so here the rest of .

The fact that this remainder of the division was given its own name is due to its importance for the so-called test of nine .

calculation

To the nines residue of a natural number to identify it first calculates the decimal checksum then this number, the checksum of this checksum so , and so on until the iterated checksum one will digits. If this results in 9, 9 is replaced by 0, because the remainder of 9 is not 9, but 0 because of (“9 divided by 9 equals 1, remainder 0”).

This way of calculating the remainder of the nine can also be extended to negative numbers by using the relationship for the checksum

attracts. Any negative nine residues that occur can be converted into positive residues by adding 9 (several times if necessary). Thus, a generalization of the nine-remainder calculation to the set of integers can be achieved.

Examples

  • n = 5387 : q (5387) = 5 + 3 + 8 + 7 = 23; q (23) = 2 + 3 = 5. The nine remainder of 5387 is 5.
  • n = 5643 : q (5643) = 5 + 6 + 4 + 3 = 18; q (18) = 1 + 8 = 9. The nine remainder of 5643 is 0.
  • n = -418 : q (-418) = - q (418) = - (4 + 1 + 8) = -13; q (-13) = - q (13) = - (1 + 3) = -4; negative result, i.e. add 9: -4 + 9 = 5. The nine remainder of -418 is 5.
  • n = +418 : q (418) = 4 + 1 + 8 = 13; q (13) = 1 + 3 = 4. The nine remainder of +418 is 4.

properties

sentence

The rule is that a number divisible by 9 (without a remainder) is always obtained if you subtract its cross sum from a natural number :

example 1

Derivation

With the decimal number representation

a -digit natural number results in:

It is

, with ,

the -th repunit (in the decimal system ), its digits are all equal to 1.

Example 2

At is , , and . So 5 is a thousand times, 4 is a hundred times, 3 is ten times and 2 is once. Considering the sum of the digits from to stay , , and left, which obviously both individually and in the aggregate without rest through 9 is divisible:

Other value systems

The procedure described above for determining the remainder of the nine is only valid in the decimal system . For other place value systems there is an analogous rule: Instead of 9 there is the largest digit of the system, i.e. the base of the place value system reduced by 1. In the hexadecimal system , the calculation is therefore with F 16 (= decimal 15), in the octal system with 7 8 . One speaks then of the hexadecimal “ F remainder ” or 15's remainder or the octal 7's remainder .

Examples in the hexadecimal system

  • n = AD37E9 : q (AD37E9) = A + D + 3 + 7 + E + 9 = 38; q (38) = 3 + 8 = B. The hexadecimal "F-remainder" (also called 15's remainder) of AD37E9 is equal to B.
  • n = 210F84 : q (210F84) = 2 + 1 + 0 + F + 8 + 4 = 1E; q (1E) = 1 + E = F; F becomes 0. The hexadecimal “F remainder” of AD37E9 is equal to 0.

Examples in the octal system

  • n = 17365 : q (17365) = 1 + 7 + 3 + 6 + 5 = 26; q (26) = 2 + 6 = 10; q (10) = 1 + 0 = 1. The octal 7's remainder of 17365 is equal to 1.
  • n = 52016734 : q (52016734) = 5 + 2 + 0 + 1 + 6 + 7 + 3 + 4 = 34; q (34) = 3 + 4 = 7; 7 becomes 0. The octal 7's remainder of 52016734 is equal to 0.

See also