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29 Jan 2020, 03:00
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
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Question Stats:
73% (01:42) correct
27%
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wrong
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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
A. 3
B. 5
C. 11
D. 14
E. 19
29 Jan 2020, 04:19
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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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=> \(10^p\) 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
General Discussion
29 Jan 2020, 04:14
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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.
