REPORTING MATH

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GROUP 2

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GROUP 2

ANDREA

AILEEN

EZEKIEL

JULLIANE

GROUP 2

ANDREA

AILEEN

EZEKIEL

JULLIANE

GROUP 2

ZYNA

MINERVA

INDEX

Modular Arithmetic

Least Residue

Modular Arithmetic

Expression involving modular arithmetic

Congruence

What is Modular Arithmetic?

Let's find out!

MODULAR ARITHMETIC

-Is a system of arithmetic for integers, which considers the remainder. -In modular arithmetic, numbers "wrap around" upon reaching a given fixed quantity (this given quantity is known as the modulus) to leave a remainder.

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WHAT IS CONGRUENCE?

Example:

>Let a and b are integers and m is a natural counting number. >a IS CONGRUENT TO b MODULO m” a ≡ b (mod m), IF m DIVIDES a - b OR b - a

Verify if the congruence is true 4 ≡ 9(mod 2)

CONGRUENCE

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EXPLANATION, LISTEN ATTENTIVELY

MORE EXAMPLES:

1. 3 ≡ 9 (mod 2)2. 4 ≡ 15 (mod 3)3. 7 ≡ 15 (mod 4) 4. 5 ≡ 20 (mod 3) 5. 4 ≡ 20 (mod 3)

It is true since 9 – 3 = 6, which is divisible by 2.

It is false since 15 – 4 = 11, which is not divisible by 3.

It is true since 15 – 7 = 8, which is divisible by 4.

It is true since 20 – 5 = 15, which is divisible by 3.

It is false since 20 – 4 = 16, which is not divisible by 3.

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What is Least Residue?

Find the Least Residue

r= 54(mod 7)

>To determine the least residue is to simply get the remainder when b is divided by m. >b(mod m) means b divided by m. m is referred to as the modulus (divisor)

Least Residue

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LEAST RESIDUE EXPLANATION

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Sample : Finding the least Residue

1. Answer is 2, since 2 is the remainder of 29/3.2. Answer is 3, since 3 is the remainder of 35/4.3. Answer is 1, since 1 is the remainder of 50/7. 4. Answer is 2, since 2 is the remainder of 30/4. 5. Answer is 0, since there is no remainder of 25/5.

1. 29 (mod 3)2.35 (mod 4) 3. 50 (mod 7) 4. 30 (mod 4) 5. 25 (mod 5)

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OPERATIONS IN MODULAR ARITHMETIC

Modulo addition is defined as (a+b)mod m

Modulo subtraction is defined as (a-b)mod m

Modulo multiplication is defined as (a*b) mod m

Modulo division is defined as (a/b) mod m

solving expression involving modular arithmetic

Solving expressions involving Modular Arithmetic

10 - 4 (mod 5) --> 1 11 + 7 (mod 3) --> 0 8 + 7 (mod 7) --> 1 20 - 7 (mod 5) --> 3 31 - 6 (mod 4) --> 1

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GROUP 2