If an integer p is to be chosen at random from 1 to 96 (both inclusive), what is the probability that p(p+1)(p+2) will be divisible by 8?

62.5%

Step-by-step explanation:

p is either even or row.

IF P IS EVEN

If even then p(p+1)(p+2) will be always be divisible by 8. p and p+2 are even numbers an at least one of those two numbers is divisible by 4.

let say that p/2=a and (p+2)/2=b, then a is even or b is b is even (b=a+1, therefore one of those two numbers must be even), so if b is divisible by 2 then (p+2) is divisible by 4, or if a is divisible by 2 then p is divisible by 4.

p(p+1)(p+2) = 2*a*(p+1)*2*b

since a or b are divisible by 2 then p(p+1)(p+2) is divisible by 8

IF P IS ROW

If p is row, then p+2 is also row, and p(p+2) is also row (and row numbers are never divisible by 8), so p(p+1)(p+2) is divisible if and only if (p+1) is divisible by 8. The list of p between 1 and 96 where (p+1) is divisible by 8 is: 7,15,23,31,39,47,55,63,71,79,87,95. A total of 12 numbers.

IN CONCLUSION

for all even numbers, between 1 and 96, p(p+1)(p+2) is divisible by 8. there are 43 even numbers.

For all the row numbers, between  1 and 96, just (p+1) divisible by 8 will be divisible by 8 . There are 12 numbers like that.

50*10 is greater

step-by-step explanation:

if you add the zero from 10 to 50, it will give you a bigger number.

- "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."