1 modulo 9 is the set of numbers as follows:
(0(9) + 1), (1(9) + 1, ....)
which is:
1, 10, 19
All we have to do is divide 1000 by 9, and round down.
1000/9 = 111.111......
There are 111 numbers.
Now let us check our work.
the set is as follows:
(1,10,19, ....... (111)9 +1)
(1, 10, 19 ........ 1000)
We are within 1000.
Therefore, 111 of the 1000 smallest positive integers are congruent to 1 modulo 9.
Answer:
Step by step explanation:
![\text{Given that, two roots are}~ -4~ \text{and}~ 4i.\\\\\text{Let,}\\\\~~~~~~~x = 4i\\\\\implies x^2 = 16i^2~~~~~~~;[\text{Square on both sides}]\\\\\implies x^2 = -16~~~~~~~~;[i^2 = -1]\\\\\implies x^2 +16 = 0\\\\\text{So,}~ x^2 +16~ \text{ is a factor of the 3 degree polynomial}.\\ \\ \text{The polynomial is ,}\\\\ f(x) = (x+4)(x^2 +16)\\\\~~~~~~~=x^3 +16x +4x^2 +64\\\\~~~~~~~=x^3 +4x^2 +16x +64](https://tex.z-dn.net/?f=%5Ctext%7BGiven%20that%2C%20%20two%20roots%20are%7D~%20-4~%20%5Ctext%7Band%7D~%20%204i.%5C%5C%5C%5C%5Ctext%7BLet%2C%7D%5C%5C%5C%5C~~~~~~~x%20%3D%204i%5C%5C%5C%5C%5Cimplies%20%20x%5E2%20%3D%2016i%5E2~~~~~~~%3B%5B%5Ctext%7BSquare%20on%20both%20sides%7D%5D%5C%5C%5C%5C%5Cimplies%20x%5E2%20%3D%20-16~~~~~~~~%3B%5Bi%5E2%20%3D%20-1%5D%5C%5C%5C%5C%5Cimplies%20x%5E2%20%2B16%20%3D%200%5C%5C%5C%5C%5Ctext%7BSo%2C%7D~%20x%5E2%20%2B16~%20%5Ctext%7B%20is%20a%20factor%20of%20the%203%20degree%20polynomial%7D.%5C%5C%20%5C%5C%20%5Ctext%7BThe%20polynomial%20is%20%2C%7D%5C%5C%5C%5C%20f%28x%29%20%3D%20%28x%2B4%29%28x%5E2%20%2B16%29%5C%5C%5C%5C~~~~~~~%3Dx%5E3%20%2B16x%20%2B4x%5E2%20%2B64%5C%5C%5C%5C~~~~~~~%3Dx%5E3%20%2B4x%5E2%20%2B16x%20%2B64)
Answer:
Jack needs to run at a speed of 1/12 mph
Step-by-step explanation:
Mathematically;
speed = distance/ time
Distance = speed * time
10 minutes = 10/60 h = 1/6 hours
The distance to the school will be
Distance = 3 * 1/6 = 1/2 = 0.5 miles
Now, he needs to be at school 4 minutes earlier, time will now be 10 - 4 = 6 minutes
To get the speed here;
Recall that the distance from above does not change
So speed = 0.5/6 mph 0r 1/12 mph
Answer: 8 friends
Step-by-step explanation:
35(at first)-3(left)=32
32(given away) divided by 4(to each friend)= 8 friends
