Thus, the greatest multiple of 5 satisfying $ u^3 < 1500 $ is $ u = 10 $.

Understanding Why $ u = 10 $ Is the Greatest Multiple of 5 with $ u^3 < 1500 $

When solving mathematical problems involving multiples and inequalities, clarity and precision are key. One common question that arises is: What is the greatest multiple of 5 such that $ u^3 < 1500 $? The answer is $ u = 10 $. But how do we determine this decisively—and why is 10 the conclusive solution?

The Math Behind the Question

Understanding the Context

We seek the largest number $ u $ that meets two conditions:

$ u $ is a multiple of 5 ($ u = 5k $ for some integer $ k $)
$ u^3 < 1500 $

Cubes increase rapidly, so only small values need testing. Let’s evaluate perfect cube roots near 1500:

$ 10^3 = 1000 $ ✅
$ 15^3 = 3375 $ ❌ (already exceeds 1500)
Try $ u = 5 $: $ 5^3 = 125 $ ✅
Try $ u = 10 $: $ 10^3 = 1000 $ ✅
Try $ u = 11, 12, 13, 14 $ — none are multiples of 5
The multiple of 5 just below 15 is 10, and $ 10^3 = 1000 $ clearly satisfies $ u^3 < 1500 $

Why All Other Multiples of 5 Fail

Thus, the greatest multiple of 5 satisfying $ u^3 < 1500 $ is $ u = 10 $.

Key Insights

Checking next higher multiples:

$ u = 15 $: $ 15^3 = 3375 > 1500 $ → invalid
Higher multiples like 20, 25, etc., produce cubes far exceeding 1500 due to exponential growth.

Thus, $ u = 10 $ is not just a candidate—it’s the largest valid multiple of 5 within the cubic bound.

Why This Problem Matters

Understanding such constraints helps in problem-solving across fields like engineering, computer science, and data modeling, where identifying feasible values under strict parameters is crucial. The reasoning illustrates how factoring smartly (noticing multiples and testing cubes) optimizes efficiency and accuracy.

Conclusion

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Final Thoughts

While many values satisfy $ u^3 < 1500 $, only $ u = 10 $ qualifies both as a multiple of 5 and a cube under 1500. Confirming this through direct computation and logical exclusion of higher multiples solidifies $ u = 10 $ as the definitive answer.

👉 Tip: When working with inequalities involving powers and multiples, test cubes systematically and use factor exclusion to cut down possibilities—this streamlines finding exact solutions.

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