For each integer n ≥ 4, let an denote
the base-n number 0:133n. The product
a4a5 : : : a99 can be expressed as m/n!
where m and n are positive integers and n
is as small as possible. What is the value
of m?
962
To solve this problem, let's first convert the base-n number 0:133n into decimal form.
In base-n, the number 0:133n can be expressed as 0 + 1/n + 3/n^2 + 3/n^3 + n/n^4.
To convert this into decimal form, we substitute each power of n with the base. For example, when n = 10, n^2 becomes 100, n^3 becomes 1000, and so on.
Now, we can calculate the decimal form of the number for a specific value of n. Let's take n = 10 for simplicity:
0:13310 = 0 + 1/10 + 3/100 + 3/1000 + 10/10000
= 0.1 + 0.03 + 0.003 + 0.0003 + 0.0001
Simplifying this expression, we get:
0.1 + 0.03 + 0.003 + 0.0003 + 0.0001 = 0.13343
Now that we have the decimal form of the number, we need to find the product a4a5...a99.
a4 = 0.13344
a5 = 0.13345
...
a99 = 0.133499
To find the product a4a5...a99, we multiply all these values together:
a4a5...a99 = (0.13344) * (0.13345) * ... * (0.133499)
However, calculating this product directly would be extremely tedious and time-consuming.
Instead, we can simplify the expression by factoring out common terms:
a4a5...a99 = 0.1334 * (0.13345) * ... * (0.133499)
Notice that the common factor is 0.1334.
There are a total of 96 terms in the product (from a4 to a99).
So the product a4a5...a99 can be written as:
a4a5...a99 = (0.1334)^96 * [(0.13345) * ... * (0.133499)]
Now, we can express the product as m/n!
m = (0.1334)^96
n = 96
Therefore, the value of m is (0.1334)^96, which is the product of 0.1334 raised to the power of 96.