In each case show your work and justify your answers. (a) If 9k+5 and 2k+1 have the same last digit for some natural number k, find the last digit of k. (b) If S = ∑ 19 i=1 i!, then find the last digit of S 2 . (c) Denote the last digit of a natural number a by b. Show that the last digit of a n is the same as the last digit of b n where is n ≥ 1 is a natural number. (d) Inspired by part (c), show that the last digit of a 4k+1 for all natural numbers k is the same as the last digit of a. [Euler’s Theorem is not allowed.]

x = 17

step-by-step explanation:

in the given figure, two triangles are inscribed in a circle where ∠bae ≅ ∠dae which makes the side be and de equal so we can write it as:

3x - 24 = x + 10

solving this equation for x to get:

3x - x = 10 + 24

2x = 34

x = 34/2

x = 17

therefore, the value of x = 17.

i you goofed up when writing this, because hnot is not a word minusf is not either. i cannot answer   because of that.

step-by-step explanation:

n=9

step-by-step explanation:

7 times a number = 7n

decreased by 13 = 7n-13

same as is =

5 times the number = 5n

increased by 5 = 5n+5

7n-13 = 5n+5

2n=18

n=9