Miss smith is organizing a field trip for her science club. She discovers that if she arranged a student into groups of four, there will be three students left over.Similarly, if she arranges them into groups of seven, there will also be three students left over. The total number of students is between 80 to 90. How many students are there in the science club?
Let's assume the total number of students in the science club is x.
According to the given condition:
x ≡ 3 (mod 4) and x ≡ 3 (mod 7)
To find the possible total number of students in the science club within the range of 80 to 90, we can start by finding the least common multiple of 4 and 7.
The least common multiple (LCM) of 4 and 7 is 28.
Now, we can find the smallest number within the range of 80 to 90 that satisfies the above condition by adding 28 to the smaller number within the range (80) until we find a number that satisfies both congruences:
80 + 28 = 108 (not within the range)
84 + 28 = 112 (not within the range)
88 + 28 = 116 (not within the range)
92 + 28 = 120 (within the range)
Therefore, the total number of students in the science club is 120.
So, there are 120 students in the science club.