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If p is a positive integer, for what minimum value of p is 7*10^p + 4

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If p is a positive integer, for what minimum value of p is 7*10^p + 4  [#permalink]

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New post 29 Jan 2020, 02:00
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If p is a positive integer, for what minimum value of p is 7*10^p + 4  [#permalink]

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New post 29 Jan 2020, 03:14
If p is a positive integer, for what minimum value of p is \(7∗10^{p}+4p\) divisible by 9?

\(7∗10^{p}+4p= 7∗(9+1)^{p}+ 4p= 7∗(9^{p}+ 9^{p-1}+...+1)+4p= ( 7∗9^{p}+ 7∗9^{p-1}+...+7 )+ 4p\)

Now, we need to check 7+ 4p whether it is divisible by 9.

A) if p=3, then 7+4∗3= 19 (not divisible by 9)
B) if p=5, then 7+4∗5= 27 (divisible by 9) -Correct

The answer is B.
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If p is a positive integer, for what minimum value of p is 7*10^p + 4  [#permalink]

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New post 29 Jan 2020, 03:19
We know that 10 divided by 9 leaves remainder 1
=> \(10^p\) divided by 9 leaves remainder 1 as well
(Any positive integer power of 10, if is divided by 9, the remainder is 1)

=> \(7 * 10^p\) divided by 9 leaves remainder 7 * 1 = 7

Since \(7 * 10^p + 4p\) is divisible by 9, we have:

The remainder when \(7 * 10^p + 4p\) is divided by 9, the remainder is 0

=> The remainder when 4p is divided by 9 must be 2
(Since \(7 * 10^p\) has remainder 7; and 2+7= 9, i.e. remainder is 0)

At p = 5, i.e. 4p = 20, the required remainder becomes 2.

Answer B

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If p is a positive integer, for what minimum value of p is 7*10^p + 4  [#permalink]

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New post 29 Jan 2020, 03:28
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Bunuel 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


For an integer to be divisible by 9, the sum of digits of the integer should be divisible by 9

A. 3
\(7 * 10^p + 4p\) = 7000+12=7012

7+0+1+2 = 10 - Not divisible by 9

B. 5
\(7 * 10^p + 4p\) = 700000+20=700020

7+0+0+0+2+0 = 9 - Divisible by 9

Answer is (B)
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Re: If p is a positive integer, for what minimum value of p is 7*10^p + 4  [#permalink]

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New post 29 Jan 2020, 07:07
Bunuel 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


divisibility rule of 9 ; sum of digits has to be 9
in this case P has to be 5 ;
IMO B
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Re: If p is a positive integer, for what minimum value of p is 7*10^p + 4  [#permalink]

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New post 02 Feb 2020, 11:13
Bunuel 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 the sum of the digits of x 10^p + 4p to be a multiple of 9. Since we are looking for a minimum value of p, let’s check the values in the given choices from the smallest to the largest.

A. 3

7 x 10^3 + 4(3) = 7000 + 12 = 7012

The sum of the digits is 10, which is not a multiple of 9, so A is not the correct answer.

B. 5

7 x 10^5 + 4(5) = 700,000 + 20 = 700,020

Since the sum of the digits is 9, B is the correct answer.

Answer: B


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If p is a positive integer, for what minimum value of p is 7*10^p + 4  [#permalink]

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New post 30 May 2020, 20:50
Bunuel 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


The fastest way to solve this problem in under 1 minute is to test the answers. For this, we will need to know some basic rules of divisibility. A number is dividisble by 9 if the SUM of its DIGITS is divisible by 9.

Since we are asked to find the minimum value, I will start by testing the lowest numbers first.

Let's try option A.
7 * 10^3 + 4*3 = 7000 + 12
The sum of the DIGITS is 10 (7+1+2)
10 is not divisible by 9.
Let's move on to the next answer option.

Let's try option B.
7 * 10^5 + 4*5 = 700,000 + 20
The sum of the DIGITS is 9 (7+2)
9 is obviously divisible by 9.

Correct answer!

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If p is a positive integer, for what minimum value of p is 7*10^p + 4   [#permalink] 30 May 2020, 20:50

If p is a positive integer, for what minimum value of p is 7*10^p + 4

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