Question
(10) Write 17! as the difference of two integer perfect squares.
Solution
4
(184 Votes)
Thaddeus
Elite ยท Tutor for 8 years
Answer
###
Explanation
## Step 1: Expressing Factorial as a Product### We can express
as
.## Step 2: Finding the Middle Term### To express
as the difference of two squares, we need to find two numbers whose product is
. We can rewrite
as the product of two equal halves. Since 17! has an odd number of factors, we can split it around the middle term. The middle term is the average of the first and last factors. The average of 1 and 17 is
. We can group the factors around 9.## Step 3: Grouping Factors### We can group the factors as follows:
. We can rewrite this as
.## Step 4: Expressing as a Difference of Squares### Let
. Then
. We want to express
as
. Let
and
. Then
and
. So,
and
. Then
.## Step 5: Calculating N###
.## Step 6: Calculating a and b###
and
.## Step 7: Verifying the Result### Since
and
are not integers, this approach doesn't work. We need to find another way to express
as a difference of squares. Let
. We can write
. Let
and
.## Step 8: Final Calculation###
and
. Then
.