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.

- "runner b is moving at a slower rate."

- "the graphs shown are graphs of distance(km) against time(min).

the slopes of this graphs are are given by:

slope = (change in y)/(change in x)

= (change in distance)/(change in time)

this is the formula for finding the speed. so, the slope of these graphs represent the speed of the runner.

taking each division in the graph to represent 1 on both axis:

runner a ⇒ speed = (2-0)/(1-0)

= 2/1

= 2 km/min

runner b ⇒ speed = (3-0)/(3-0)

= 3/3

= 1 km/min

this shows that   runner b is moving at a slower rate."

it would open into a square depending on the type of envelope i’m not sure about the second part