Find the least positive integer which gives remainder 1 when divided by 13, and remainder 2 when divided by 31.

The required integer is 157.

Step-by-step explanation:

Let the unknown number be x.

It is given that the remainder 1 when x is divided by 13, and remainder 2 when x is divided by 31.

Here,

13 and 31 hare prime numbers. so the GCD o 13 and 31 is

Using Chinese remainder theorem, we get

The formula to find the value of x is

Substitute the given values in the above formula.

Let n=-1, because we need to find the least positive integer.

Therefore, the required integer is 157.

b. 153 students do not participate in sports and seventy-five 12th graders were surveyed

a

step-by-step explanation:

the denominator cannot be the same as one of the factors.

8 denominator times 2 denominator = 16

and 7 numerator time 11 = 77

77/16= 77/16 ( they cannot be further simplified)