MathDB

5

Part of 2024 Mathematical Talent Reward Programme

Problems(2)

MTRP OBJECTIVE Q5

Source: MTRP 2024

3/16/2024
How many positive integers nn between 11 and 20242024 (both included) are there such that ⌊n⌋\lfloor{\sqrt{n}}\rfloor divides nn? (For x∈R,⌊n⌋x \in \mathbb{R}, \lfloor{n}\rfloor denotes the greatest integer less than or equal to xx.) (A)44(A) 44 (B)132(B) 132 (C)1012(C) 1012 (D)2024(D) 2024
number theory
MTRP SUBJECTIVE Q5

Source: MTRP 2024

3/16/2024
Let f:N⟶Nf:\mathbb{N} \longrightarrow \mathbb{N} such that f(m)−f(n)=f(m−n)10n∀m>n∈Nf(m) - f(n) = f(m-n)10^n \forall m>n \in \mathbb{N}. Additionally, gcd(f(k),f(k+1))=1∀k∈N(f(k),f(k+1)) = 1 \forall k \in \mathbb{N}. Show that if a,ba,b are coprime natural numbers, that is, gcd(a,b)=1(a,b) = 1 then f(a),f(b)f(a),f(b) are also coprime.
algebra