1. A positive integer $n$ is called divisibly prime if for every positive divisor $d$ of $n$, at least one of the numbers $d-1$ and $d+1$ is a prime number. For example, 8 is divisibly prime because its positive divisors 1, 2, 4, and 8 each differ by 1 from a prime number (namely 2, 3, 5, and 7, respectively), while 9 is not divisibly prime because its divisor 9 does not differ by 1 from a prime number (both 8 and 10 are composite).

(a) Which odd numbers can be a divisor of a divisibly prime number?

(b) Determine the largest divisibly prime number.
## 1. Version for Grade 5 & Grade 4 and below

(a) Suppose $n$ is a semi-prime number. Then $n$ cannot have an odd divisor $d \geqslant 5$. Indeed, for such a divisor, both $d-1$ and $d+1$ are even numbers. Since $d-1>2$, both are composite numbers, which would contradict the fact that $n$ is semi-prime. The odd divisors 1 and 3 can occur, because the number 3 is itself semi-prime.

(b) Due to the unique factorization into prime factors, the number $n$ can only have factors of 2 and at most one factor of 3. The number $2^{6}=64$ and all multiples thereof are not semi-prime, because neither $63=7 \cdot 9$ nor $65=5 \cdot 13$ is a prime number. A semi-prime number can therefore have at most five factors of 2. The largest number that could possibly be semi-prime is thus $3 \cdot 2^{5}=96$.

We check that 96 is indeed a semi-prime number: the divisors of 96 are 1, 2, 3, 4, 6, $8,12,16,24,32,48$ and 96, and the neighboring numbers $2,3,2,3,5,7,11,17,23,31,47$ and 97 are prime. The largest semi-prime number is therefore 96.