To prove that a number's square cannot end with two consecutive odd digits, let's consider the following:

Step 1: Represent the number

Let $n$ be any integer, and we want to prove that $n^2$ (the square of $n$) cannot end with two consecutive odd digits.

Step 2: Analyze the last digit of $n$

We know that the last digit of $n^2$ depends only on the last digit of $n$. Thus, we can reduce the problem by examining the possible last digits of $n$, which are $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$, and compute the corresponding last two digits of $n^2$.

Step 3: Calculate last two digits of $n^2$

For each $n \mod 10$, compute $n^2 \mod 100$ (i.e., the last two digits of the square of $n$):

1. $n = 0 \Rightarrow n^2 = 0^2 = 00$
2. $n = 1 \Rightarrow n^2 = 1^2 = 01$
3. $n = 2 \Rightarrow n^2 = 2^2 = 04$
4. $n = 3 \Rightarrow n^2 = 3^2 = 09$
5. $n = 4 \Rightarrow n^2 = 4^2 = 16$
6. $n = 5 \Rightarrow n^2 = 5^2 = 25$
7. $n = 6 \Rightarrow n^2 = 6^2 = 36$
8. $n = 7 \Rightarrow n^2 = 7^2 = 49$
9. $n = 8 \Rightarrow n^2 = 8^2 = 64$
10. $n = 9 \Rightarrow n^2 = 9^2 = 81$

Step 4: Check the last two digits

Now let's list the last two digits of each square:

• $0^2 = 00$
• $1^2 = 01$
• $2^2 = 04$
• $3^2 = 09$
• $4^2 = 16$
• $5^2 = 25$
• $6^2 = 36$
• $7^2 = 49$
• $8^2 = 64$
• $9^2 = 81$

Step 5: Conclusion

From this list, we see that no square of a number ends with two consecutive odd digits. The possible last digits of squares are: $00, 01, 04, 09, 16, 25, 36, 49, 64, 81$. None of these end with two consecutive odd digits (such as 11, 33, 55, 77, or 99).

Thus, we have proved that a number's square cannot end with two consecutive odd digits.

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Would you like any more details on this proof or any part clarified?

Related questions:

1. Why can't a square of a number end with 77?
2. How do the last two digits of squares follow specific patterns?
3. Can a square of a number end with two even digits?
4. What other properties can be derived from the last digits of a number's square?
5. How can this property be generalized for bases other than 10?

Tip:

When solving number theory problems, exploring modular arithmetic is a powerful tool to detect patterns efficiently.