Four positive integers $A$, $B$, $C$ and $D$ have a sum of 64. If $A+3 = B-3 = C \times 3 = D \div 3$, what is the value of the product $A \times B \times C \times D$?
We have that $A + B + C + D = 64$.  Substituting everything in terms of $C$, we find that $(3C - 3) + (3C + 3) + C + (9C) = 64$, which means that $C = 4$.  Thus $A = 9$, $B = 15$, and $D = 36$.  Therefore our desired answer is $9\cdot 15\cdot 4\cdot 36 = \boxed{19440}$.
19440
Level 4
Algebra
Problem: Four positive integers $A$, $B$, $C$ and $D$ have a sum of 64. If $A+3 = B-3 = C \times 3 = D \div 3$, what is the value of the product $A \times B \times C \times D$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>Let the common value of the expressions \( A + 3 \), \( B - 3 \), \( C \times 3 \), and \( D \div 3 \) be denoted as \( x \). Then we can express \( A \), \( B \), \( C \), and \( D \) in terms of \( x \):

1. From \( A + 3 = x \), we derive \( A = x - 3 \).
2. From \( B - 3 = x \), we derive \( B = x + 3 \).
3. From \( C \times 3 = x \), we derive \( C = \frac{x}{3} \).
4. From \( D \div 3 = x \), we derive \( D = 3x \).

Next, we substitute these expressions into the equation for the sum of the integers:

\[
A + B + C + D = (x - 3) + (x + 3) + \frac{x}{3} + 3x = 64.
\]

Simplifying this, we combine like terms:

\[
x - 3 + x + 3 + \frac{x}{3} + 3x = 5x + \frac{x}{3}.
\]

To combine \( 5x \) and \( \frac{x}{3} \), we convert \( 5x \) into a fraction:

\[
5x = \frac{15x}{3}.
\]

Thus, we have:

\[
\frac{15x}{3} + \frac{x}{3} = \frac{16x}{3}.
\]

Setting this equal to 64 gives:

\[
\frac{16x}{3} = 64.
\]

To solve for \( x \), we multiply both sides by 3:

\[
16x = 192.
\]

Dividing by 16 yields:

\[
x = 12.
\]

Now we can find the values of \( A \), \( B \), \( C \), and \( D \):

1. \( A = x - 3 = 12 - 3 = 9 \).
2. \( B = x + 3 = 12 + 3 = 15 \).
3. \( C = \frac{x}{3} = \frac{12}{3} = 4 \).
4. \( D = 3x = 3 \times 12 = 36 \).

Next, we calculate the product \( A \times B \times C \times D \):

\[
A \times B \times C \times D = 9 \times 15 \times 4 \times 36.
\]

Calculating step-by-step:

1. \( 9 \times 15 = 135 \).
2. \( 4 \times 36 = 144 \).
3. Now, \( 135 \times 144 \).

To compute \( 135 \times 144 \):

\[
135 \times 144 = 135 \times (100 + 40 + 4) = 13500 + 5400 + 540 = 19440.
\]

Thus, the final answer is \( 19440 \).<|END_OF_LATENT|> The final answer is $19440$.