Question

If s and t are positive integers such that \(\frac{s}{t} = 64.12\), which of the following could be the remainder when s is divided by t?

• A. 2
• B. 4
• C. 8
• D. 20
• E. 45

Hint:

When you see an equation like \(\frac{s}{t} = \text{decimal}\), immediately separate the decimal into its integer and fractional parts. The integer part is the quotient, and the fractional part is the remainder divided by the divisor (\(r/t\)).

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The division of integer s by integer t can be expressed as \(s = qt + r\), where q is the quotient and r is the remainder. Dividing this equation by t gives \(\frac{s}{t} = q + \frac{r}{t}\). We can use this relationship to find the remainder.
Step 2: Key Formula or Approach:
From the given equation: \[ \frac{s}{t} = 64.12 = 64 + 0.12 \] Comparing this to \(\frac{s}{t} = q + \frac{r}{t}\), we can identify: The integer part is the quotient, \(q = 64\).
The fractional part is the ratio of the remainder to the divisor, \(\frac{r}{t} = 0.12\).
Step 3: Detailed Explanation:
We have the equation for the remainder r and the divisor t: \[ \frac{r}{t} = 0.12 \] Let's convert the decimal to a fraction and simplify it: \[ \frac{r}{t} = \frac{12}{100} = \frac{3}{25} \] This gives us the relationship \(25r = 3t\).
Since r and t must be integers, this equation tells us that r must be a multiple of 3, and t must be a multiple of 25.
Now we check the given options to see which one could be a value for r. The value must be a multiple of 3. (A) 2 is not a multiple of 3.
(B) 4 is not a multiple of 3.
(C) 8 is not a multiple of 3.
(D) 20 is not a multiple of 3.
(E) 45 is a multiple of 3 (\(45 = 3 \times 15\)).
Let's verify if \(r=45\) is possible. If \(r=45\), then \(t = \frac{25r}{3} = \frac{25 \times 45}{3} = 25 \times 15 = 375\). The condition for a remainder is \(0 \leq r<t\). Here, \(0 \leq 45<375\), which is true. Thus, 45 is a possible remainder.
Step 4: Final Answer:
The only possible value for the remainder among the choices is 45.