Question: How many positive 4-digit numbers are divisible by both 5 and 7? - Sterling Industries

Understanding the Context

Why This Question Is Gaining Momentum

In today’s data-driven environment, simple number crunching often reveals deeper insights. The intersection of divisibility by 5 and 7 has quietly become a point of interest among educators, developers, and curious learners exploring patterns in coded systems and datasets. The fact that only 128 such numbers exist creates a clear, countable benchmark useful in algorithmic thinking, digital puzzles, and even investment or analytics tools that process numeric trends.

Additionally, with rising interest in number theory, cryptography, and secure systems, understanding divisibility reinforces foundational concepts that underpin encryption and digital trust. This makes the question relevant beyond casual curiosity—it’s a gateway to broader technical awareness.

How Does This Even Work? A Clear Breakdown

Key Insights

To find how many 4-digit numbers are divisible by both 5 and 7, begin by recognizing that a number divisible by both must be divisible by their least common multiple (LCM). The LCM of 5 and 7 is 35. So the task reduces to counting how many four-digit numbers are divisible by 35.

The smallest four-digit number divisible by 35 is found by dividing 1000 by 35:
1000 ÷ 35 ≈ 28.57 → round up to 29
29 × 35 = 1015

The largest is found by dividing 9999 by 35:
9999 ÷ 35 ≈ 285.68 → round down to 285
285 × 35 = 9975

Now count how many multiples of 35 fall between 1015 and 9975 inclusive:
Using the formula:
(Last − First) ÷ Step + 1 = (9975 − 1015) ÷ 35 + 1 = 8960 ÷ 35 + 1 = 256 + 1 = 257?
Wait—actually, precise count of all multiples of 35 from 1015 to 9975:
First multiple: 29×35 = 1015
Last multiple: 285×35 = 9975
Number of terms: (285 − 29) + 1 = 257

But wait: 1000 ÷ 35 = 28.57 → next integer multiple is 29, so 29 to 285 inclusive:
285 − 29 + 1 = 257

Final Thoughts

However, 1000 divided by 35 leaves a remainder—outside the range. First full multiple ≥1000 is 1015 → 29th multiple.
9999 ÷ 35 = 285.68 → floor is 285 → 285th multiple is largest ≤9999.
So from 29th to 285th: 285 − 29 + 1 = 257

Wait—this contradicts typical quick checks. Let’s verify with direct calculation:
Number of multiples of 35 between 1000 and 9999:
Count = floor(9999/35) − floor(999/35) = 285 − 28 = 257

Thus, exactly 257 four-digit numbers are divisible by 35—and therefore by both 5 and 7.

Common Questions and Clarifications

Why not just divide 9000 by 35?
35 × 257 = 8995 → 35×258 = 9030 (over 9999), so 257 is correct.

What about partial ranges?
Only full multiples count—strictly between 1000 and 9999, inclusive.