How many numbers between 450 and 460 will leave a remainder when divided by all of 3 4 5 7 and 8?

Oh, dude, let's break this down. So, the numbers that leave a remainder when divided by 3, 4, 5, 7, and 8 are basically not multiples of any of those numbers. So, let's see, the multiples of these numbers are 3x4x5x7x8 = 840. So, between 450 and 460, there are 10 numbers, but only 1 of them (459) is not a multiple of 840. So, there's your answer, just one number between 450 and 460 fits the bill.