Find A in abcd=A(four digital number) is a perfect square ,given ab=2cd+1

In summary, the formula for finding a perfect square in the given equation is ab=2cd+1, where a and b are two digits and c and d are consecutive digits. To determine if a given number is a perfect square, solve the equation and check if the resulting value of a is a perfect square number. Any four-digit number can be used for this equation, as long as it follows the format of abcd. The possible values for a, b, c, and d in this equation are integers, and a and b must satisfy the equation ab=2cd+1. This equation is specific to finding perfect squares in four-digit numbers and cannot be used for numbers with a different number of digits.
  • #1
Albert1
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$\overline{abcd}=A$(four digital nmber) is a perfect square ,given $\overline{ab}=2\overline{cd}+1$
find $A=?$
 
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  • #2
My solution:

We may state:

\(\displaystyle (20c+2d+1)100+10c+d=n^2\)

or:

\(\displaystyle 67(30c+3d)=(n+10)(n-10)\)

Let's let $n=77$ so we have:

\(\displaystyle 67(30c+3d)=87\cdot67\)

Hence:

\(\displaystyle 30c+3d=87\)

\(\displaystyle 10c+d=29=10\cdot2+9\)

Thus we see we have:

\(\displaystyle c=2,\,d=9\implies a=5,\,b=9\)

And so:

\(\displaystyle A=5929=77^2\)
 
  • #3
My solution:

Let $x=\overline{ab}$, $y=\overline{cd}$, and $A=n^2$.

Then:
$$A=100x+y = n^2 \land x=2y+1 \quad\Rightarrow\quad
100(2y+1)+y=n^2 \quad\Rightarrow\quad
3\cdot 67 \cdot y=(n-10)(n+10)
$$
Since $67$ is prime, it must divide either $n-10$ or $n+10$.
Since $n$ must be a 2-digit number, we conclude:
$$n-10=67 \lor n+10 = 67 \quad\Rightarrow\quad n=57 \land n=77$$
Only $A=77^2=5929$ satisfies the condition, so that is the one and only solution. (Smile)
 

FAQ: Find A in abcd=A(four digital number) is a perfect square ,given ab=2cd+1

What is the formula for finding a perfect square in the given equation?

The formula for finding a perfect square in the given equation is ab=2cd+1, where a and b are two digits and c and d are consecutive digits.

How do I know if the given number is a perfect square?

You can determine if the given number is a perfect square by solving the equation ab=2cd+1 and checking if the resulting value of a is a perfect square number.

Can I use any four-digit number for this equation?

Yes, you can use any four-digit number for this equation. However, the number must follow the format of abcd, where a and b are two digits and c and d are consecutive digits.

What are the possible values of a, b, c, and d in this equation?

The possible values for a, b, c, and d in this equation are integers, where a and b are two digits and c and d are consecutive digits. Additionally, a and b must satisfy the equation ab=2cd+1.

Can I use this equation to find perfect squares with more or less than four digits?

No, this equation is specifically for finding perfect squares in four-digit numbers. For numbers with a different number of digits, a different equation or method would need to be used.

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