Find the greatest 4-digit number divisible by 15, 25, 40, and 75.

Given data

• Divisibility by all of: 15, 25, 40, 75.

Concept / Approach

• The required numbers are multiples of L.C.M.(15, 25, 40, 75).
• Find the largest multiple ≤ 9999.

Step-by-step calculation

Prime forms: 15 = 3 × 5; 25 = 5^2; 40 = 2^3 × 5; 75 = 3 × 5^2.LCM = 2^3 × 3 × 5^2 = 8 × 3 × 25 = 600.Largest 4-digit multiple of 600: ⌊9999 ÷ 600⌋ = 16 ⇒ 16 × 600 = 9600.

Verification

9600÷15, 9600÷25, 9600÷40, 9600÷75 are all integers.

Common pitfalls

• Treating it as G.C.D. problem instead of L.C.M.
• Rounding 9999/600 incorrectly (it must be the floor).

Final Answer

Greatest such number: 9600.