nkmungila wrote:

If \(p\) is a positive integer, for what minimum value of \(p\) is \(7 * 10^p + 4p\) divisible by 9?

A. 3

B. 5

C. 11

D. 14

E. 19

We need a multiple of 9. So the number's nonzero digits must total 9 or a multiple of 9.

The key is the term +\(4p\)

7 * 10 to ANY power (70; 7,000; 7,000,000) leaves one nonzero integer: 7

7 + 4p's digits must = 9 / multiple of 9

So \(4p\) must, e.g., equal 2 or 11 or 20 or 38. . . to yield a multiple of 9, thus:

7 + 2 = 9

7 + 3 + 8 = 18

\(10^{p}\) does not matter.

Answer choices, checking only to see if 4p + 7 = multiple of 9

A) p = 3

(4)(3) = 12

(1 + 2 + 7) = 10, and (1 + 0 = 1)

Not divisible by 9. NO

Number = \(7 * 10^3 + (4)(3)= 7,012\)

B) p = 5

(4)(5) = 20

7 + 2 + 0 = 9

That works. YES

Number = \(7 * 10^5 + (4)(5)= 70,020\)

C. 11. (7 + 4*11) = 51. NO

D. 14. (7 + 4*14) = 63. YES

E. 19. (7 + 4*19) = 83. NO

Answers B and D work. But the prompt asks for the minimum value of p: 5 < 14

Answer B