A book with 53 pages numbered 1 to 53 has its pages renumbered in reverse, from 53 to 1. For how many pages do the new page number and old page number share the same units digit?
Each page is assigned two numbers. We can generalize the numbers assigned to page $x$ to be the pair $x$ and $54-x$ for $1 \leq x \leq 53$. That is, if $x = 1$, then we can see that page number one is assigned the numbers $1$ and $54-1 = 53$. It is fairly easy to see that the units digits of $x$ and $54-x$ will only be the same if $x$ has a units digit of $2$ or a units digit of $7$. Thus, we simply must count how many such $x$ there are between $1$ and $54$. Possibilities for $x$ are: 2, 7, 12, 17, 22, 27, 32, 37, 42, 47, and 52. Therefore, there are $\boxed{11}$ such pages.
11
Level 5
Prealgebra
Problem: A book with 53 pages numbered 1 to 53 has its pages renumbered in reverse, from 53 to 1. For how many pages do the new page number and old page number share the same units digit?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine how many pages share the same units digit after renumbering, we first identify the old and new page numbers. The old page numbers range from 1 to 53, while the new page numbers range from 53 to 1. 

Next, we denote the old page number as \( x \) and the new page number as \( y \). The relationship between \( x \) and \( y \) is given by the equation \( y = 54 - x \). 

We need to find the values of \( x \) for which the units digits of \( x \) and \( y \) are the same. This can be expressed mathematically as:

\[
x \mod 10 = y \mod 10
\]

Substituting \( y \) into the equation gives:

\[
x \mod 10 = (54 - x) \mod 10
\]

This simplifies to:

\[
x \mod 10 = (4 - x \mod 10) \mod 10
\]

Let \( d = x \mod 10 \). The equation becomes:

\[
d = (4 - d) \mod 10
\]

This leads to two cases based on the properties of modular arithmetic:

1. If \( 4 - d \geq 0 \), then \( d = 4 - d \) which simplifies to \( 2d = 4 \) or \( d = 2 \).
2. If \( 4 - d < 0 \), then \( d = 4 - d + 10 \) which simplifies to \( 2d = 14 \) or \( d = 7 \).

Thus, the possible units digits for \( x \) are 2 and 7. 

Next, we find the old page numbers that correspond to these units digits:

- For \( d = 2 \): The old page numbers are 2, 12, 22, 32, 42. This gives us 5 pages.
- For \( d = 7 \): The old page numbers are 7, 17, 27, 37, 47. This also gives us 5 pages.

Adding these counts together, we find that the total number of pages where the new and old page numbers share the same units digit is:

\[
5 + 5 = 10
\]

However, we must also consider the page numbered 52, which has a units digit of 2 and corresponds to the new page number 2. This brings the total to:

\[
10 + 1 = 11
\]

Thus, the final answer is 11.<|END_OF_LATENT|> The final answer is $11$.