Grades 9-10 Video Solutions 2013-Level 9&10 Video Solutions 2013 problem21

Video Transcription

Question number 21, the number 1 divided by 1,024,000 is written as a decimal with the smallest possible number of digits. How many digits appear after the decimal point?  So first, let's write down what it is that we're looking at. We have a decimal representation. We're not sure how many digits, but we can write  down something that looks like 0, a, b, c, d for the digits.  There will be, let's say, n digits, and that is going to be equal to the number we are given.  So 1 divided by 1,024,000. Now, on the left here of the equal sign, we want to find out exactly the smallest value of n, and so we can do  that algorithmically, sort of recursively, if you will,  by taking this number and multiplying it by 10 to some value n until we do not have digits after the decimal on the left-hand side, so until we come up with an integer.  So let's see what happens when we factor this. So we can factor, and what we obtain is in the numerator,  2 to the power n times 5 to the power n,  and in the denominator we will have the factorization 1,024 times 1,000, so 2 to the power 3 times 5 to the power 3, 10 cubed is 1,000, 2 to the power 10 is 1,024,  and so that gives us 2 to the power n minus 13 times 5 to the power n minus 3, and this number is an integer so long as we choose n bigger than 13.  But we want the smallest value of n, and that corresponds exactly to the number of digits after the decimal, which is 13.

Video Summary

The problem involves determining how many digits are needed after the decimal point for the number \(1 \div 1,024,000\) to be expressed with the smallest possible number of digits. By factoring \(1,024,000\) as \(2^{10} \times 5^3\) and applying this to our expression, we establish the condition that the minimum number of digits \(n\) after the decimal will result when \(n - 3 = 10\), because the factors of 2 and 5 must be balanced to form an integer. This gives \(n = 13\), so there are 13 digits after the decimal point.

Keywords

decimal digits

division

factorization

integer expression

minimum digits