https://file.aiquickdraw.com/mathbot/file/1725903691945pq3181w8.jpg

Let's solve each part of the problem by finding the smallest whole number that satisfies each congruence.

### Part (a)
We need to find the smallest whole number \( x \) such that:
\[
-7 \equiv x \pmod{6}
\]

To solve this, we need to find the remainder when \(-7\) is divided by \(6\). We can add \(6\) repeatedly until the result is non-negative:

\[
-7 + 6 = -1
\]

\[
-1 + 6 = 5
\]

So, \(-7 \equiv 5 \pmod{6}\). The smallest whole number for \(x\) that makes the statement true is **5**.

### Part (b)
We need to find the smallest whole number \( y \) such that:
\[
10 - 17 \equiv y \pmod{9}
\]

First, compute \(10 - 17 = -7\). Now, we need to find the remainder when \(-7\) is divided by \(9\). Similar to part (a), we add \(9\) repeatedly until the result is non-negative:

\[
-7 + 9 = 2
\]

So, \(-7 \equiv 2 \pmod{9}\). The smallest whole number for \(y\) that makes the statement true is **2**.

### Final Answers:
- (a) \(x = 5\)
- (b) \(y = 2\)

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. What is the general method for solving linear congruences?
2. How can we solve a system of congruences using the Chinese Remainder Theorem?
3. What are some real-world applications of modular arithmetic?
4. How does modular arithmetic apply to cryptography?
5. How can we find the multiplicative inverse in modular arithmetic?

**Tip:** When working with modular arithmetic, always remember that adding or subtracting multiples of the modulus does not change the congruence.

For each part below, fill in the blank with the smallest whole number that makes the statement true.
(a) -7 ≡ □ (mod 6)
(b) 10 - 17 ≡ □ (mod 9)

Solve modular arithmetic problems where you need to find the smallest whole numbers that satisfy the given congruences. Learn to compute remainders and understand modular arithmetic with step-by-step solutions. Suitable for students in Grades 9-12.

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