CMI BS Hons Chapter-wise PYQs

Because the value of $f$ on any positive integer is determined by its values on primes. Also, so For part (a), factor the given numbers: Let Then Also And So for all integers built only from primes $2,3,5$, the value is uniquely determined. Hence The first positive integer whose value is not determined is the first prime not among $2,3,5$, namely So the smallest such $n$ is For part (b), yes, such a function exists. Choose any prime Define the exponent of $p$ in the prime factorization of $n$. Then so this does satisfy Now if then $p$ cannot divide $n$, so Thus $f(x)=0$ for every positive integer $x<2025^{2025}$, but so $f$ is not identically zero. For part (c), first note that Therefore by Bézout''s identity there exist integers $u,v,w$ such that Define Since $a,b,c$ are positive rationals and $u,v,w$ are integers, $r_0$ is again a positive rational number. Using the additive property, So at least one such rational exists. Now we show there are infinitely many. Choose two distinct primes $p,q$. Since the codomain is the integers, the numbers $g(p)$ and $g(q)$ are integers. Consider This is a positive rational number, and If $s\neq 1$, then for every integer $t$, The numbers $r_0s^t$ are all distinct, so this already gives infinitely many rationals with value $1$. If $s=1$, then unique factorization forces and then with $\frac{p}{q}\neq 1$. Again for all integers $t$, giving infinitely many distinct rational numbers. Hence there are infinitely many positive rational numbers $r$ such that The relevance of the codomain being the integers is exactly that exponents such as $g(p)$ and $g(q)$ are integers, so the constructed kernel element is again a rational number.