There is a prime number P. if we add it to 12.18.24, it is still a prime number

There is a prime number P. if we add it to 12.18.24, it is still a prime number

There is a prime number P. if you add it to 12.18.24, it is still a prime number. The prime number is 5
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Prime number 10 = () + () + () 18 = () * () * ()

10=（2）+（3）+（5）18=（2）*（3）*（3）

Only prime numbers can be filled in 18 = () + () 37 = () + () + () 60 + () + ()

18=（7）+（11） 37=（11）+（7）+（19） 60=（41）+（17）+（2）

6 = () * () prime 5 = () + () 18 = () + () 21 = () * () 30 = () * () * () all prime numbers

6=2*3
5=2+3
18=11+7
21=3*7
30=2*3*5

Find the surface area of the plane X / A + Y / B + Z / C = 1 in the first trigram limit part, hope to be more detailed, thank you

This figure is to take a line segment of length a, B and C on the X, y and Z axes respectively, and then form a tetrahedron. S (total) = 1 / 2 (AB + BC + Ca) + s (inclined triangle)
S (inclined triangle) can be solved by Helen's formula

Use a 28 decimeter long wire to form a rectangle. The ratio of the length to width of the rectangle is 4:3. How many square meters is the area of the rectangle?
Please

28 △ 2 = 14 decimeters
Length: 14 × 4 / (4 + 3) = 8 decimeters
Width: 14-8 = 6 decimeters
Area: 8 × 6 = 48 square decimeters

Given that x2 + 2x + 5 is a factor of X4 + AX2 + B, find the value of a + B

Let X4 + AX2 + B = (x2 + 2x + 5) (x2 + MX + n) = X4 + (2 + m) X3 + (2m + N + 5) x2 + (5m + 2n) x + 5N. By comparing the coefficients of corresponding terms, we get 2 + M = 02m + N + 5 = a5m + 2n = 05n = B. the solutions are m = - 2, n = 5, a = 6, B = 25 ∥ a + B = 31

Circle C: x ^ 2 + y ^ 2-4x-14y + 45 = 0 and point Q (- 2,3)
(1) P (a, a + 1) on the circle, find the length of the line PQ and the slope of the line PQ
(2) Find the length of the chord ab of the line PQ satisfying (1) cut by the circle C
(3) If M is any point of the circle C, find the maximum and minimum of | MQ |
(4) If the real number m, n satisfies m ^ 2 + n ^ 2-4m-14n + 45 = 0, find the maximum and minimum value of k = (n-3) / (M + 2)

(1) Take (a, a + 1) into the circle equation and get a = 4
So the point P coordinates are (4,5)
PQ = root sign [(4 + 2) ^ 2 + (5-3) ^ 2] = 2 root sign 10
(3) The maximum value of MQ | is the distance d from Q to the center of the circle plus the radius of the circle; the minimum value of MQ | is the distance d from Q to the center of the circle minus the radius of the circle
x²+y²-4x-14y+45=0
(X-2) & sup2; + (Y-7) & sup2; = (2 radical 2) & sup2;
The center coordinates of the circle are (2,7), and the radius of the circle is 2
The distance from Q to the center of the circle is: D = radical [(- 2-2) & sup2; + (3-7) & sup2;] = 4 radical 2
So the maximum value of | MQ | is: 4 radical 2 + 2 radical 2 = 6 radical 2
|The minimum value of MQ | is: 4 radical 2-2 radical 2 = 2 radical 2
(4) K is the slope of the line
If there are two tangent lines, take the larger one; (the smaller one is the smallest straight line.)
x^2+y^2-4x-14y+45=0 ①
y-3=k(x+2) ②
It's time to eliminate y
(k^2+1)x^2+4(k^2-2k-1)x+4(k^2-4k+3)=0
There are two identical roots
16(k^2-2k-1)^2-16(k^2+1)(k^2-4k+3)=0
Simplify
k^2-4k+1=0
Take the larger root
K (max) = 2 + radical 3
K (min) = 2-radical 3

If x is an integer so that | 4x ^ 2-12x-27 | is a prime number, what values can x take? Please explain why

X is an integer, | 4x ^ 2-12x-27 | = | (2x-9) (2x + 3) | is a prime number
2x-9 = ± 1 or 2x + 3 = ± 1
x=5
x=4
x=-1
x=-2
The above four solutions are in line with the meaning of the question

When the value of X is, the absolute value of x minus 2 has a minimum value. What is the minimum value?

The minimum value of x = 2 is 0