Here we will explain what 3 mod 109 means and show how to calculate it. 3 mod 109 is short for 3 modulo 109 and it can also be called 3 modulus 109.
Modulo is the operation of finding the Remainder when you divide two numbers. Therefore, when you ask "What is 3 mod 109?" you are asking "What is the Remainder when you divide 3 by 109?".
We will show you two methods of finding 3 mod 109 (3 modulo 109). To differentiate our methods, we will call them the "Modulo Method" and the "Modulus Method".
Before we continue, we remind you of what the different parts of a division problem are called so you can follow along: Dividend / Divisor = Quotient and in this case 3 is the Dividend, 109 is the Divisor, and the answer is called the Quotient.
Furthermore, the Quotient x.y has two parts: x to the left of the decimal point is the Whole part, and y to the right of the decimal point is the Fractional part.
Modulo Method
To find 3 mod 109 using the Modulo Method, we first divide the Dividend (3) by the Divisor (109).
Second, we multiply the Whole part of the Quotient in the previous step by the Divisor (109).
Then finally, we subtract the answer in the second step from the Dividend (3) to get the answer. Here is the math to illustrate how to get 3 mod 109 using our Modulo Method:
3 ÷ 109 ≈ 0.027523
0 × 109 = 0
3 - 0 = 3
Thus, the answer to "What is 3 mod 109?" is 3.
Modulus Method
To find 3 mod 109 using the Modulus Method, we first find the highest multiple of the Divisor (109) that is equal to or less than the Dividend (3).
Then, we subtract the highest Divisor multiple from the Dividend to get the answer to 3 modulus 109 (3 mod 109):
Multiples of 109 are 0, 109, 218, 327, etc. and the highest multiple of 109 equal to or less than 3 is 0. Therefore, to get the answer:
3 - 0 = 3
Thus, once again, the answer to "What is 3 mod 109?" is 3.
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