Which among the following is the smallest 7 digit number that is exactly divisible by 43?

1. Translate the problem requirements: We need the smallest 7-digit number that is divisible by 43. The smallest 7-digit number is \(1,000,000\), but this may not be divisible by 43, so we need to find the next number after \(1,000,000\) that is exactly divisible by 43.
2. Find the remainder when dividing the smallest 7-digit number by 43: Calculate what's left over when \(1,000,000\) is divided by 43 to understand how far we are from the next multiple.
3. Calculate the adjustment needed: If there's a remainder, determine how much we need to add to \(1,000,000\) to reach the next multiple of 43.
4. Verify against answer choices: Check which answer choice matches our calculated result and confirm it's divisible by 43.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we're looking for. We need the smallest 7-digit number that is exactly divisible by 43.

The smallest 7-digit number is \(1,000,000\) (one followed by six zeros). However, this number might not be divisible by 43. If it's not divisible by 43, we need to find the next number larger than \(1,000,000\) that IS divisible by 43.

Think of it like this: imagine you're looking for parking spaces numbered in multiples of 43, and you need the first space that's numbered \(1,000,000\) or higher. If space \(1,000,000\) isn't a multiple of 43, you need to walk forward until you find the next multiple.

Process Skill: TRANSLATE - Converting the word problem into a clear mathematical task

2. Find the remainder when dividing the smallest 7-digit number by 43

Now we need to see what happens when we divide \(1,000,000\) by 43.

Let's do this division step by step:
\(1,000,000 \div 43 = ?\)

To find this, I'll calculate: \(1,000,000 \div 43 = 23,255\) with a remainder.

Let me verify: \(23,255 \times 43 = 999,965\)

So when we divide \(1,000,000\) by 43:

• We get 23,255 complete groups of 43
• This accounts for 999,965
• The remainder is: \(1,000,000 - 999,965 = 35\)

This means \(1,000,000 = 43 \times 23,255 + 35\)

The remainder is 35, which tells us that \(1,000,000\) is 35 more than the nearest smaller multiple of 43.

3. Calculate the adjustment needed

Since \(1,000,000\) has a remainder of 35 when divided by 43, we need to add something to get to the next multiple of 43.

Think about it this way: if we've gone 35 steps past a multiple of 43, how many more steps do we need to reach the NEXT multiple of 43?

We need: \(43 - 35 = 8\) more steps.

So the smallest 7-digit number divisible by 43 is:
\(1,000,000 + 8 = 1,000,008\)

Let's verify: The next multiple of 43 after 999,965 would be:
\(999,965 + 43 = 1,000,008\) ✓

4. Verify against answer choices

Looking at our answer choices:

1. 1,000,043
2. 1,000,008 ← This matches our calculation!
3. 1,000,006
4. 1,000,002
5. 1,000,001

Let's double-check that \(1,000,008\) is divisible by 43:
\(1,000,008 \div 43 = 23,256\) exactly (no remainder)

Verification: \(23,256 \times 43 = 1,000,008\) ✓

The answer is B. 1,000,008.

Common Faltering Points

Errors while devising the approach

Students might confuse themselves about what exactly is the smallest 7-digit number. Some may think it's 1,111,111 or get confused about counting digits properly. The smallest 7-digit number is definitively \(1,000,000\) (one followed by six zeros), but this foundational step needs to be crystal clear.

The key insight is understanding that we need to find the smallest multiple of 43 that is greater than or equal to \(1,000,000\). Students might try to approach this by testing each answer choice individually rather than using the systematic approach of finding the remainder and calculating the adjustment needed.

Errors while executing the approach

When dividing \(1,000,000\) by 43, students are prone to calculation errors. The division \(1,000,000 \div 43 = 23,255\) remainder 35 involves multiple steps, and any mistake in the multiplication (\(23,255 \times 43 = 999,965\)) or subtraction (\(1,000,000 - 999,965 = 35\)) will lead to an incorrect remainder.

Once students find that \(1,000,000\) leaves a remainder of 35 when divided by 43, they need to calculate \(43 - 35 = 8\) to find how much to add. Some students mistakenly think they should subtract the remainder instead of finding the complement, or they might add the remainder itself (35) instead of the complement (8).

Errors while selecting the answer

After calculating \(1,000,000 + 8 = 1,000,008\), students should verify that this number is actually divisible by 43. Skipping this verification step means missing the chance to catch any earlier calculation errors. Students should confirm that \(1,000,008 \div 43 = 23,256\) with no remainder.