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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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A. 3
B. 5
C. 11
D. 14
E. 19
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sujoykrdatta
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Tutor
Joined: 26 Jun 2014
Last visit: 23 Aug 2026
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Expert reply
GMAT 1: 750 Q51 V39
Posts: 598
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
Posted from my mobile device
_________________
=> \(10^p\) 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
Posted from my mobile device
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General Discussion
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.
\(7∗10^{p}+4p= 7∗(9+1)^{p}+ 4p= 7∗(9^{p}+ 9^{p-1}+...+1)+4p= ( 7∗9^{p}+ 7∗9^{p-1}+...+7 )+ 4p\)
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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Bunuel
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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Bunuel
divisibility rule of 9 ; sum of digits has to be 9
in this case P has to be 5 ;
IMO B
ScottTargetTestPrep
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Expert reply
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Bunuel
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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Bunuel
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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Bunuel
\(7 * 10^p\) will result in 70........... (No of 0's will depend on the value of p and the same is irrelevant as sum will be 7 irrespective of the value of p)
Thus, for \(7 * 10^p + 4p\) to be divisible by 9 the value of \(7 * 10^p + 4p\) must be divisible by 9...
Again we know, the sum of \(7 * 10\) is 7...
So, The deciding factor is the value of \(4p\)
Now, check the options...
(A) \(7*10^3 + 4*3 = 7+12 =19\) (Not divisible by 9)
(B) \(7*10^5 + 4*5 = 7+20 = 27\) (Divisible by 9) (Answer)
If you want you can check and reconfirm the option...
_________________
why not 14 then ? 7+5+6 = 18
firas92
bcecharly
If p = 14, 7 * 10^p + 4p is alos divisible by 9 but it's not the minimum possible value of p:
If p is a positive integer, for what minimum value of p is \(7 * 10^p + 4p\) divisible by 9?
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