To determine the lowest power of 10 by which you can multiply the divisor and dividend to yield a whole number, we need to consider the following:
Assume we have a fraction \(\frac{a}{b}\), where \(a\) is the dividend and \(b\) is the divisor. If we want to make the divisor \(b\) into a whole number, we need to find the lowest power of 10, say \(10^n\), such that both \(a \cdot 10^n\) and \(b \cdot 10^n\) are integers.
The key is recognizing that if \(b\) is not already an integer, it may be a fraction. Generally, \(b\) can be expressed in its simplest form as \(\frac{p}{q}\), where \(p\) is an integer, and \(q\) is the greatest integer:
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If \(b\) is already an integer, then \(n = 0\) (since multiplying by \(10^0 = 1\) gives no change).
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If \(b\) is not an integer and expressed as a fraction \(\frac{p}{q}\):
- The lowest power of 10 \(10^n\) that makes \(b \cdot 10^n\) an integer would be when \(n\) equals the number of trailing zeros in the denominator when written in simplified form.
For example:
- If \(b = \frac{1}{2}\), \(10^1 = 10\) will convert it to \(10 \cdot \frac{1}{2} = 5\) (whole number).
- If \(b = \frac{5}{8}\), you'd need to multiply by \(10^3\) since it's in the simplest form with a denominator that cannot be simplified further with powers of 10 directly, making it \(1000 \cdot \frac{5}{8} = 625\) (whole number).
Thus, in general, the answer to the problem can be framed as follows:
- The lowest power of 10 you multiply by is based on ensuring the denominator is converted into a whole number through the decimal places in \(b\).
- The power required \(n\) corresponds to the number of decimal places in \(b\) to clear fractions.
In conclusion, we denote this \(n = \text{The number of trailing zeros required to express } b \text{ as a whole number.}\)
This concept can be visualized as increasing the denominator of each allowed fraction until it becomes a whole number by shifting decimal points efficiently, hence arriving at a whole number fraction.