JEEOlympiad
A positive integer is called 'rough' if it is divisible by the square…
Question
A positive integer is called 'rough' if it is divisible by the square of some prime (equivalently, it is not squarefree). Find the smallest positive integer n such that all three of n, n+1, n+2 are rough.
✓ Verified answer: 48checked by our engine — not a guess
Step-by-step solution
We need three consecutive non-squarefree integers.
Check small triples.
A number is non-squarefree iff 4|n, 9|n, 25|n, 49|n, ...
for some prime square.
Scanning upward: 48 = 16·3 (divisible by 4), 49 = 7^2, 50 = 2·25 (divisible by 25).
All three are non-squarefree.
One verifies no smaller n works (e.g.
8,9 are non-squarefree but 10 is squarefree; 24,25 then 26 squarefree; etc.), so the first run of three is 48,49,50.
Final answer48
Stuck on a problem like this?
Paste any JEE or NEET question — verified working, a confidence %, and an honest “not sure” instead of a bluff.
Solve my doubt →More Number Theory solutions
Find the remainder when the power tower 7^(7^7) is divided by 1000.…JEE · MathHow many trailing zeros does 2026! have when it is written in base…JEE · MathLet D be the set of all positive divisors of 2025. Compute the sum of…JEE · MathFind the number of ordered pairs of positive integers (a, b) such…JEE · MathConsider repdigit numbers made entirely of the digit 8, i.e. 8, 88,…JEE · MathLet sigma(n) denote the sum of all positive divisors of n. Count the…JEE · MathFind the number of integers x with 0 ≤ x < 18000 satisfying x^2 ≡ 1…JEE · MathCompute the value of the sum S = sum_{k=1}^{100} k · gcd(k, 100).JEE · Math