# Largest prime number with +, -, ÷

This is inspired by this great puzzle

What is the largest prime number that can be made using the integers from 1 to 10 under the following conditions:

• Addition, subtraction, division are allowed.
• Multiplication and other operations are not allowed
• All numbers must be used exactly once
• Last condicition is superfluous. What about concatenation - is it allowed?
– z100
Commented Sep 4 at 21:25

I believe the answer will be

$$907,199$$

obtained through

$$10 \div (2 \div 3 \div 4 \div 5 \div 6 \div 7 \div 8 \div 9) - 1 = \frac{10!}{4}-1 = 907,199$$

Note that

adding 1 at the end instead of subtracting doesn't yield a prime number, as $$907,201 = 53 \times 17117$$

• Nice try as 10/(1/2/4/5/6/7/8/9)+3=1209603 is not a prime. But I think something better may be achieved creating two values very close to each other and subtract them. .
– z100
Commented Sep 4 at 21:48
• I figured out pretty quickly that removing a 2, 3, or 4 wouldn't work because the remaining factorial would be divisible by what you've removed due to having 8 or 9 in it. I don't think the method you've described will be able to net a very high number, but you're welcome to try. Commented Sep 4 at 21:52
• Note: variations such as 9 / (2/3/4/5/6/7/8/10) - 1 achieve the same 907,199. Commented Sep 5 at 17:01
• Yeah, you can swap around the positions of all the numbers bar 1 and 2 and the sum will still be the same. I don't think it's possible to get any higher with swaps though. Commented Sep 5 at 21:13