+244 A mystery number is greater than 50 and less than 100. You can make exactly five different rectangles with the mystery number of tiles. It's prime factorization consists of only one prime number. What is the number?
Pls help me. Everyone in my class is confused!
+257 By looking at b and c together, we can deduce that 5 is the prime number in the mystery number's factorization, so we know our number is a product of 5 and one or more non-primes. Since the non-primes must be even numbers, and we know the factors can only contain ONE prime, the remaining factors MUST be multiples of 2, so our number is 5*2^x such that 5*2^x > 50 and 5*2^x < 100:
5*2*2*2*2 = 80
Our mystery number is 80!
I hope this helps!
+129852 There really is no such number......here's why.....
If the number consisted of only one prime factor....it would have to be either a square, a cube, or 3^4 = 81....
But.....the only cube that exists between 50 and 100 is 4^3 = 64 = 2^6
And the divisors of 64 are 1 | 2 | 4 | 8 | 16 | 32 | 64 ......however, we can only form 4 possible rectangles with these divisors - 1 x 64, 2 x 32, 4 x 16 or 8x 8
So......the number must be a square......but the only squares between 50 and 100 are 64 and 81 and....we have seen that 64 isn't possible
And the divsors of 81 are 1 | 3 | 9 | 27 | 81 .........and only 3 rectangles are possible......1 x 81, 3 x 27 and 9 x 9...... [note, for the same reason, 3^4 = 81 isn't possible, either ]
80 would provdie us with 5 rectangles because its dvisors are 1 | 2 | 4 | 5 | 8 | 10 | 16 | 20 | 40 | 80
And the rectangles are : 1 x 80, 2 x 40, 4, 20, 5 x 16 and 8 x 10
But 80 has two prime factors [ 2 and 5] because 80 = 2^4 x 5
So.....no number exactly matches the given criteria......
