4 Remainder Arithmetic and Divisibility
1 Remainder Arithmetic Modulo 10
2 Remainder Arithmetic Modulo 10 more
3 Remainder Arithmetic Modulos 10 more still
4 Remainder Arithmetic Modulo 10 in general
5 Remainder Arithmetic Modulo 5
6 Remainder Arithmetic Modulo 5 Properties
7 Remainder Arithmetic Modulo 5 Examples I
8 Remainder Arithmetic Morulo 5 Examples II
9 Remainder Arithmetic Divisibility by 5
10 Remainder Arithmetic Long Division by 5 Quickly
11 Remainder Arithmetic Long Division by 5 Quickly more
12 Remainder Arithmetic Modulo 10 Example
13 Remainder Arithmetic Modulo 5 Example
14 Remainder Arithmetic Modulo 9 Example
15 Remainder Arithmetic Modulo 9 Example
16 Remainder Arithmetic Modulo 9 Example 2
17 Remainder Arithmetic Rule of 9 for checking sums I
18 Remainder Arithmetic Rule of 9 for checking sums II
19 Remainder Arithmetic Rule of 9 for checking sums III
20 Remainder Arithmetic Rule of 9 for checking sums IV
21 Remainder Arithmetic Modulo 3
22 Remainder Arithmetic Modulo 3 more
23 Remainder Arithmetic Modulo 2
24 Divisibility Tests for 2 3 5 9 10
25 Divisibility Tests for 2 3 5 9 10 Examples
26 Divisibility by 2 3 5 Examples
27 Divisibility by 2 3 6 5 9 10 Examples
A Decimals Modular and Remainder Arithmetic
Notes
The following lessons provide a very slow, essentially prealgebraic
introduction to remainder arithmetic for modulo 10, 5, 9, 3 and 2 to to
explain the origin of divisibility rules for division by 2, 3, 6, 9 and
10. The lessons start with remainder arithmetic modulo 10 as decimals are
based on counting in groups or powers of 10.
-
Remainder Arithmetic Modulo 10 calculates the remainders on
division by 10 for two whole numbers 5430 and 253, and then asks what
the remaindeer on division by 10 will be for their sums and products.
-
Remainder Arithmetic Modulo 10 more calculates the remainders
on division by 10 for whole numbers 846134 and 25683, and then shows
what the remainder on division by 10 will be for their sum.
-
Remainder Arithmetic Modulo 10 still more calculates the
remainders on division by 10 for whole numbers 846134 and 25683, and
then shows what the remainder on division by 10 will be for their
product
-
Remainder Arithmetic Modulo 10 in general uses the patterns
that arose in the previous lessons or examples to calculate
remainders, modulo 10, for many sums and products.
-
Remainder Arithmetic Modulo 5 calculates the remainders or
values, modulo 5, of whole numbers 1 to 10. There-in lies an
introduction to Remainder Arithmetic, modulo 5.
-
Remainder Arithmetic Modulo 5, Properties describes the
properties in an algebraic way, and then gives an example to
illustrate the product property.
-
Remainder Arithmetic Modulo 5 Examples I calculates the
remainders, modulo 5 for 10, 230, 345, 348 and 347. The remainder for
347 is left as 7, modulo 5. The latter equals 2, modulo 5.
-
Remainder Arithmetic Modulo 5 Examples II calculates the
remainders, modulo 10 for 14583 and from the latter, the remainder
modulo 5 for a 14583. The exercise is repeated for 84767 = 7 modulo
10 or 2 modulo 5. The conclusion is that each whole number its last
digit modulo 5.
-
Remainder Arithmetic - Divisibility by 5. The foregoing
discussion of remainder arithmetic, modulo 5, sets the stage for
recognizing divisibility by 5. There in lies a justification of the
divisibility rule - a whole number is divisible by 5 when the last
digit of its decimal representation is a 0 or a 5. Note the number
nineteen has a decimal numeral representation 19 with a last digit 9,
but the Roman numeral or reprsentation is XIX, a representation to
which the divisibility rules does not apply.
-
Remainder Arithmetic - Long Division by 5 Quickly. For each
whole number, division by 5 can be done by long - or short - division
methods. Here is alternative to both based on the decimal
representation of the whole number.
-
Remainder Arithmetic - Long Division by 5 Quickly more. The
alternative in the previous lesson is shown again for another
example.
-
Remainder Arithmetic Modulo 10 Example gives an arithmetic
expression 56 + 13×28 and calculates the remainder modulo
10 for it.
-
Remainder Arithmetic Modulo 5 Example takes t
expression 56 + 13×28 and calculates the remainder modulo
5 and modulo 9 for it. This example should be read after the next
lesson on Remainder arithmetic, modulo 9.
-
Remainder Arithmetic Modulo 9 Example . This lesson
show 10, 100, 1000 and 10000 all have remainder 1, modulo 9. Then it calculates
8362 modulo 9.
-
Remainder Arithmetic Modulo 9 Example /b>. Calculate
537, Modulo 9
-
Remainder Arithmetic Modulo 9 Example 2. Illustration of the sum of
digits rule for calculating remainders, modulo 9.
-
Remainder Arithmetic Rule of 9 for checking sums, Example I illustrated for
the sum of 537, 821 and 674.
-
Remainder Arithmetic Rule of 9 for checking sums, Example II illustrated for
the sum of 824, 560, 301 and 24.
-
Remainder Arithmetic Rule of 9 for checking sums, Example III illustrated for
the sum of 421, 321, 567 and 222.
-
Remainder Arithmetic Rule of 9 for checking sums IV illustrate for
sum of 44375 and 82621.
-
Remainder Arithmetic Modulo 3 shows that 10, 100, 1000 and 10000 are all
equal to 1, modulo 3. Then calculates 8432 modulo 3 from its decimal expansion
in terms of 10, 100, and 1000 - a sum of digit rule results.
-
Remainder Arithmetic Modulo 3 more - the question of when is a
number a multiple of 3, and if not what is the remainder. Remainder
arithmetic calculation, modulo 3, is applied to 4856.
-
Remainder Arithmetic Modulo 2 calculates remainders on division by
2 for the whole numbers 1 to 10. Then calculates the remainder modulo 2,
for whole numbers 3453, 5674, and 47, and arrives at a last digit rule.
-
Divisibility Tests for 2 3 5 9 10 - Each number has a decimal form.
Based on that form, we have rules for recognizing when a whole number is
divisible by when of the divisors 2, 3, 5, 9 and 10.
-
Divisibility Tests for 2 3 5 9 10 Examples - which are the numbers
45, 55, 31, 4 and 369 are multiples of 2, 3, 5, 9 10. Divisibility rules
and remainder arithmetic are applied.
-
Divisibility by 2 3 5 Examples. Test numbers 560, 382,
184, 962, 866 and 118 for divisibility by 2, 3 or 5.
-
Divisibility by 2, 3, 6, 5, 9 , 10 and 100 Questions. Calculate the remainders
of the numbers of the numbers 242, 1500, 173, 180, 37 and 600 on division by
2, 3, 6, 5, 9 , 10 and 100
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