Carcass wrote:
OE
Quote:
Because the only even prime number is 2, m must be the product of 2 × 2 = 4 and two odd primes. Further, due to the restriction that m be less than 300, the product of those two odd primes must be less than \(\frac{300}{4} = 75\). Thus, you need to find the largest number smaller than 75 that is the product of exactly two odd prime numbers. You can rule out even numbers, and you have 73, which is prime, as is 71; 69 is the product of two prime numbers (3 and 23). Thus, the maximum possible value for m is 2 × 2 × 3 × 23 = 276.
A slightly faster was to get to 69. Once you know that you need to find the largest number smaller than 75 that is the product of exactly two odd prime numbers, it's fair to guess one of the prime factors is 3.* What is the other prime factor? The largest prime number smaller than 75/3, aka largest prime < 25. That's 23. 23*3 is 69.
Multiply 69 by 4, and you get the answer.
*This is just an intuition I have, but I checked it against other possibilities and it seemed to be true. Like, if I guessed the prime factor was 5 or 13, that wouldn't work. Could someone explain why 3 is a reasonable guess here? My guess is something to do with, you want to find a number whose product is close to 75, and generally the product of two numbers far apart on the number line is going to be more than the product of numbers closer? But that seems wrong? It seems like there's something generalizable here.