Ms Math's kindergarten class has 16 registered students. The classroom has a very large number, N, of play blocks which satisfies the conditions:
(a) If 16, 15, or 14 students are present in the class, then in each case all the blocks can be distributed in equal numbers to each student, and
(b) There are three integers 0 < x < y < z < 14 such that when x, y, or z students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.
Find the sum of the distinct prime divisors of the least possible value of N satisfying the above conditions.
N is divisible by 3*5*7*16 = 1680.
Find the appropriate multiple of 1680:
for m=1:200
n=1680*m;
howMany=0; set=[];
for i=4:13
r=mod(n,i);
if r==3
howMany=howMany+1;
set(end+1)=i;
end
end
if howMany>=3
disp([m,n,howMany])
disp(set)
end
end
The result shows
131 * 1680 = 220080 blocks would have exactly 3 integers in the given range that leave a remainder of 3. They are 9, 11, and 13.
The sum of the distinct prime factors of 220080 can be found with:
>> factor(220080)
ans =
2 2 2 2 3 5 7 131
>> unique(factor(220080))
ans =
2 3 5 7 131
>> sum(unique(factor(220080)))
ans =
148
The answer is 148.
But 220,080 is a lot of blocks for one classroom.
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Posted by Charlie
on 2024-04-23 14:49:10 |
computer-aided solution
