Question

1. The number of students in a school is between 500 and 600. If we group them into groups of 12, 20, or 36 each, 7 students are always left over. How many students are in this school?

Answer 1

If we had 77 students less, the number of students would have been divisible by 12,20,3612,20,36 separately.

The smallest number divisible by 12,20,3612,20,36 is their least common multiple which is:

   lcm(12,20,36) = 180

So the number of students will be a number of the form 180n+7. To find this n we have to use the fact that the number of students is between 500 and 600, so by trying we find that n=3.

Therefore the number of students is:  180×3 + 7 = 540 +7 = 547

.12 =2 * 2 * 3

 20 =2 * 2* 5

 36 =2 * 2 * 3 * 3

  2 * 2 * 3 * 3 * 5 =180

  2 * 2 * 3 * 3 * 5 =180

   3 * 180 = 540

   540 + 7 = 547

Answer 2

Concept Introduction:-

Fractions must be reduced to a single low common denominator (lcd) before being added, subtracted, or compared.

Given Information:-

We have been given that The number of students in a school is between $500$ and $600$. If we group them into groups of $12, 20$ or $36$ each, $7$ students are always left over.

To Find:-

We have to find that number of students are in this school

Solution:-

According to the problem

The LCM of $[12, 20, 36] = 180$, then smallest N is:

$180m + 7 =187$, where $m=0, 1, 2, 3......etc$.

Since the number of students is between $500$ and $600$ students, then:

$180 * 3 + 7 = 547$ - Number of students in this school.

Final Answer:-

The number of students in the school is $547$.