In the statement below, the two blanks can be filled by positive single-digit numbers in such a way that the statement is always true:
$$\text{If }2x\equiv y+5\ (\bmod\ 9)\text{, then }x\equiv \underline{\ \ \ }\,y+\underline{\ \ \ }\ (\bmod\ 9).$$What is the product of the two digits that go in the blanks?
Multiplying both sides of the congruence 2x≡y+5 (mod 9) by 5 gives 10x≡5y+25(mod 9),then reducing both sides modulo 9 gives x≡5y+7(mod 9).Thus, the product of the blanks is 5*7=35.
Well, let's see. To find the product of the two digits that go in the blanks, we need to find the values that satisfy the equation $2x \equiv y+5\ (\text{mod}\ 9)$ where $x \equiv a,\ b\ (\text{mod}\ 9)$.
If we consider all possible remainders when dividing by 9, we can start by looking at what happens when $x \equiv 0\ (\text{mod}\ 9)$. In this case, $2x \equiv 0\ (\text{mod}\ 9)$ and $y+5 \equiv y+5\ (\text{mod}\ 9)$. So for the equation to hold, we need $0 \equiv y+5\ (\text{mod}\ 9)$. This means $y \equiv -5\equiv 4\ (\text{mod}\ 9)$.
Now let's see what happens when $x \equiv 1\ (\text{mod}\ 9)$. In this case, $2x \equiv 2\ (\text{mod}\ 9)$ and $y+5 \equiv y+5\ (\text{mod}\ 9)$. Again, for the equation to hold, we need $2 \equiv y+5\ (\text{mod}\ 9)$. This means $y \equiv -3\equiv 6\ (\text{mod}\ 9)$.
Continuing this pattern, we find that the possible values for $y$ are 4, 6, 8, 1, 3, 5, 7, and 0. Since we're looking for positive single-digit numbers, the values of $a$ and $b$ are limited to 0, 1, 2, 3, 4, 5, 6, 7, and 8.
Now, we need to think "positive" and look for combinations that satisfy $x \equiv a,\ b\ (\text{mod}\ 9)$. After some quick calculations, we find that the only combination that works is $x \equiv 1,\ 4\ (\text{mod}\ 9)$.
Therefore, the product of the two digits that go in the blanks is $1 \times 4 = \boxed{4}$. That's the number that was "mod"-est about this problem!
To determine the value of the two missing digits, let's analyze the given statement step-by-step.
The congruence given is $$2x\equiv y+5\ (\bmod\ 9).$$
We want to determine the congruence $$x\equiv \underline{\ \ \ }\,y+\underline{\ \ \ }\ (\bmod\ 9).$$
To solve this, we can start by simplifying the given congruence. We can subtract 5 from both sides:
$$2x-5\equiv y\ (\bmod\ 9).$$
Now, if we divide both sides by 2, we have:
$$x-\frac{5}{2}\equiv \frac{y}{2}\ (\bmod\ 9).$$
Since we want our congruence to involve only integers, we need to work with integer coefficients. To achieve this, we can multiply both sides by 2:
$$2x-5\equiv 2y\ (\bmod\ 9).$$
This congruence is equivalent to:
$$2x\equiv 2y\ (\bmod\ 9).$$
Now, we can compare this congruence with the given congruence:
$$2x\equiv y+5\ (\bmod\ 9).$$
Since both congruences are equivalent, we have:
$$2y\equiv y+5\ (\bmod\ 9).$$
Subtracting $y$ from both sides gives:
$$y\equiv 5\ (\bmod\ 9).$$
Therefore, the missing values are $x\equiv 2$ and $y\equiv 5\ (\bmod\ 9)$.
The product of these two digits is $2 \times 5 = 10$.
Hence, the product of the two digits that go in the blanks is $\boxed{10}$.
To solve this problem, we need to analyze the given statement and determine the values that can be filled in the two blanks to make it true for all possible values of $x$ and $y$.
Let's start by examining the given congruence equation: $2x\equiv y+5\ (\text{mod}\ 9)$.
To simplify this equation, we can subtract $5$ from both sides: $2x-5\equiv y\ (\text{mod}\ 9)$.
Now, let's consider the equation $x\equiv ?\,y+?\ (\text{mod}\ 9)$.
We can multiply the congruence equation we obtained earlier by $4$ on both sides: $8x-20\equiv 4y\ (\text{mod}\ 9)$.
Simplifying further, we have: $8x\equiv 4y\ (\text{mod}\ 9)$.
Since multiplying an equation by a non-zero integer does not change its congruence, this means that the congruences $2x-5\equiv y\ (\text{mod}\ 9)$ and $8x\equiv 4y\ (\text{mod}\ 9)$ are equivalent.
Thus, $2x-5\equiv y\ (\text{mod}\ 9)$ is the same as $8x\equiv 4y\ (\text{mod}\ 9)$.
To make the statement true, we need to find two single-digit numbers, $a$ and $b$, such that $x\equiv a\,y+b\ (\text{mod}\ 9)$ is always true.
Comparing the coefficients on both sides, we have $8\equiv a\ (\text{mod}\ 9)$ and $4\equiv b\ (\text{mod}\ 9)$.
The value of $a$ can be $8$, $17$, $26$, etc., as long as $a$ is congruent to $8$ modulo $9$.
The value of $b$ can be $4$, $13$, $22$, etc., as long as $b$ is congruent to $4$ modulo $9$.
Since we are looking for single-digit numbers, the only two numbers that satisfy the given congruences are $a=8$ and $b=4$.
Therefore, the values that go in the blanks are $8$ and $4$, respectively.
Finally, the product of the two digits that go in the blanks is $8 \times 4 = \boxed{32}$.