A total of 1250 pens were distributed among some students such that each student received the maximum possible number of pens. What is the number of students provided that the number of pens with each student is a perfect cube?
answer) 10*5^3???
what after that
@oobleck
please help
thank you @oobleck
I think you did it
2 * 5 * 5*3
each gets 125
so there are 10 students
To find the number of students when each student is given a perfect cube number of pens, we need to find a factorization of 1250 into perfect cubes.
First, we list the perfect cubes from 1 to 1250:
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
To distribute 1250 pens among students, we need to find a combination of these perfect cubes whose sum adds up to 1250.
Starting from the largest perfect cube (1000), we check if it divides evenly into 1250. If it does, then we can distribute the pens equally and the number of students will be 1250 divided by that perfect cube.
1000 is not a factor of 1250, so we move to the next largest perfect cube, which is 729. 729 is a factor of 1250 because 1250 divided by 729 equals 1 with a remainder of 521.
Next, we need to find the number of students. Since the remainder is 521, it means that we have already distributed some of the pens to the students. We need to find the remaining number of pens that can be distributed to the students evenly.
Now, subtract 521 from 1250:
1250 - 521 = 729
We are left with 729 pens remaining to be distributed. We repeat the process with the remaining perfect cubes to find the number of students.
Next, we check 512. 512 is a factor of 729 because 729 divided by 512 equals 1 with a remainder of 217.
Subtract 217 from 729:
729 - 217 = 512
We are left with 512 pens remaining.
Next, check 343. 343 is a factor of 512 because 512 divided by 343 equals 1 with a remainder of 169.
Subtract 169 from 512:
512 - 169 = 343
We are left with 343 pens remaining.
Next, check 216. 216 is not a factor of 343, so we move to the next perfect cube.
Next, check 125. 125 is a factor of 343 because 343 divided by 125 equals 2 with a remainder of 93.
Subtract 93 from 343:
343 - 93 = 250
We are left with 250 pens remaining.
Next, check 64. 64 is not a factor of 250, so we continue.
Next, check 27. 27 is not a factor of 250, so we continue.
Next, check 8. 8 is a factor of 250 because 250 divided by 8 equals 31 with a remainder of 2.
Subtract 2 from 250:
250 - 2 = 248
We are left with 248 pens remaining.
Finally, check 1, which is always a factor of any number.
1 is a factor of 248 because 248 divided by 1 equals 248 with no remainder.
Subtract 0 from 248:
248 - 0 = 248
We are left with 248 pens.
We have now gone through all the perfect cubes and found the remainders for each of them.
The number of students will be the sum of the quotients obtained when dividing 1250 by each perfect cube, with the remainder at the end.
Therefore, the number of students is:
1 + 1 + 2 + 1
Adding the remainders:
1 + 1 + 2 + 1 + 0 = 5 students
So the answer is 5 students.