If x ≡ 4 (mod 5), which of the following values of x is NOT possible?

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Question: If x ≡ 4 (mod 5), which of the following values of x is NOT possible?

Options:

Correct Answer: 19

Solution:

x can be expressed as 4 + 5k, where k is an integer. 19 is not congruent to 4 modulo 5.

If x ≡ 4 (mod 5), which of the following values of x is NOT possible?

If x ≡ 4 (mod 5), which of the following values of x is NOT possible?

Step 1: Understand what 'x ≡ 4 (mod 5)' means. This means that when x is divided by 5, the remainder is 4.
Step 2: Write the general form of x based on the equation. We can express x as x = 4 + 5k, where k is any integer (like 0, 1, -1, etc.).
Step 3: Calculate some possible values of x by substituting different integers for k. For example, if k = 0, x = 4; if k = 1, x = 9; if k = -1, x = -1, and so on.
Step 4: Check the given values of x to see if they can be expressed in the form 4 + 5k.
Step 5: For each value, divide it by 5 and check the remainder. If the remainder is 4, then that value is possible. If not, it is NOT possible.
Step 6: Identify which value does not give a remainder of 4 when divided by 5.

Modular Arithmetic – Understanding congruences and how to express numbers in terms of a modulus.
Integer Representation – Recognizing that numbers can be represented in the form of a base plus a multiple of the modulus.