When a number is multiplied by 13 and 13 is added to the product, the resultant is divisible by 5. Find the smallest number possible?a)85b)130c)65d)90e)105Correct answer is option 'C'. Can you explain this answer?

Question

Answer

To find the smallest number that satisfies the given condition, we need to take into account the divisibility rule for 5. A number is divisible by 5 if its units digit is either 0 or 5.

Let's assume the number we are looking for is represented by 'x'. According to the given condition, when 'x' is multiplied by 13 and 13 is added to the product, the result should be divisible by 5.

So we can write the equation as:
13x + 13 ≡ 0 (mod 5)

Now, let's simplify the equation step by step:

Step 1: Subtract 13 from both sides of the equation:
13x ≡ -13 (mod 5)

Step 2: Simplify -13 (mod 5) to get the congruence in a more manageable form:
13x ≡ 2 (mod 5)

Step 3: Find the modular inverse of 13 with respect to 5:
The modular inverse of 13 (mod 5) is 2, since 13 * 2 ≡ 1 (mod 5).

Step 4: Multiply both sides of the congruence by the modular inverse of 13:
2 * 13x ≡ 2 * 2 (mod 5)
26x ≡ 4 (mod 5)

Step 5: Simplify 26 (mod 5) to get the congruence in a more manageable form:
x ≡ 4 (mod 5)

Now, we need to find the smallest number that satisfies this congruence. By trial and error, we can easily see that the smallest number that satisfies x ≡ 4 (mod 5) is 4 itself.

Therefore, the smallest number that satisfies the given condition is 4.

Since option C represents the number 65, which is not the correct answer, it seems there may be an error in the question or the options provided. Please double-check the options and the question statement.

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