ALGEBRAIC CONGRUENCE PART 1
Yes, may congruence po sa algebra. Kung sa geomtery, triangle congruence sa algebra meron din.
Also known as "Modular Arithmetic"
General form
a = b mod c
(a is congruent to b modulo c)
Normally tatlong guhit talaga sya. Pero wala kasi akong ganung symbol dito sa keyboard hahaha kaya = na lang muna pero tatlong guhit dapat.
When we say a = b mod c, the remainder when a is divided by c, is b.
Halimbawa:
7 = 1 mod 6
12 = 0 mod 12
PROPERTIES:
If a = b mod c then,
0 ≤ b ≤ c - 1
Kasi nga remainder si b, so pwede syang 0.
If a = b mod c and b > c then a = (c-b) mof c
Halimbawa
15 = 3 mod 2 = (2 - 1) mod 2 = 1 mod 2
If a = -b mod c then a = (c-b) mod c
Halimbawa:
15 = -1 mod 16
Kasi kailangan pa ng 1 si 15 para maging divisible ng 16.
29 = -1 mod 2 = (2-1) mod 2 = 1 mod 2
If a = b mod c and d = e mod c then (a±d) = (b±e) mod c
Halimbawa
5 = 2 mod 3 and 10 = 1 mod 3
Thus,
5+10 = 2+1 mod 3
15 = 3 mod 3 = 0 mod 3
If a = b mod c, and k is a positive integer, then ka = kb mod c
Halimbawa
10 = 3 mod 7, kapag nagmultiply ka ng 2 both sides,
2(10) = 2(3) mod 7
20 = 6 mod 7
If a = b mod c then a^n = b^n mod c if n is a positive integer.
Halimbawa
6 = -1 mod 7
Square ka both sides
6² = (-1)² mod 7
36 = 1 mod 7
Let us apply them to problems:
What is the remainder when 2^2018 is divided by 7?
Solution:
We know that 2³ = 8 = 1 mod 7
2018 = 3(672) + 2
So,
(2³)^672 = 1^672 mod 7
2^2016 = 1 mod 7
Multiply 2² on both sides
(2²)(2^2016) = 2² mod 7
2^2018 = 4 mod 7.
Exercise:
When x is divided by 7, the remainder is 5, what is the remainder when 9x is divided by 7?
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-Shiro
