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If ‘p’ is a positive integer, for what minimum value of ‘p’ is

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If ‘p’ is a positive integer, for what minimum value of ‘p’ is  [#permalink]

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New post 25 Oct 2017, 22:42
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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

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If ‘p’ is a positive integer, for what minimum value of ‘p’ is  [#permalink]

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New post 25 Oct 2017, 23:09
A number is divisible by 9 if the sum of all of its digits is divisible by 9.
Example- 126783 is divisible by 9 since the sum of its digits is 27, which is divisible by 9.

Here (7*10^p + 4p) can be seen as made up of the parts 7*10^p and 4p.
For any value of p, 7*10^p will only have one non zero digit and that will be 7 itself. (Followed by p number of zeroes)

4p on the other hand depends on the value of p.
We go by plugging in options. The correct answer is 5, so I’ll prove it works for that. You should do option A first.

Consider option B. p=5
4p=20
Then 700000+20 will have just 7 and 2 as non zero digits. So the sum of all the digits will be 9, which is divisible by 9.


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If ‘p’ is a positive integer, for what minimum value of ‘p’ is  [#permalink]

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New post 26 Oct 2017, 15:28
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
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Re: If ‘p’ is a positive integer, for what minimum value of ‘p’ is  [#permalink]

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New post 26 Oct 2017, 18:13
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A number is divisible by 9 when the sum of all of its digits is divisible by 9.

7*\(10^p\)= 70, 700, 7000....... (for p= 1,2,3......)

Now 7+4p should be divisible by 9

(A) 7+4*3= 19
(B) 7+4*5= 27 (Divisible by 9)

Answer: B.
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Re: If ‘p’ is a positive integer, for what minimum value of ‘p’ is  [#permalink]

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New post 26 Oct 2017, 18:23
This is little bit time saving.

When 7*10^p is divided by 9 reminder will be 7 (always). Now if (7*10^p +4p) is divisible with then 4p/9 reminder should be 2 so, addition of both reminder (7+2=9) will divisible by 9.

Now if p=3 then 4p=12 reminder (4p/9) is 3 this means at p=3 not possible here.

If p=5 then 4p=20, reminder 2 so, this is the correct Answer.

Continue with option 3 , 4 and 5.

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Re: If ‘p’ is a positive integer, for what minimum value of ‘p’ is  [#permalink]

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