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If the sum of 5 consecutive positive multiples of a prime number is 10

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If the sum of 5 consecutive positive multiples of a prime number is 10  [#permalink]

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New post 01 Jun 2017, 13:28
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If the sum of 5 consecutive positive multiples of a prime number is 105, what is the sum of the first 5 positive multiples of that prime number?


A. 30
B. 45
C. 75
D. 105
E. Can't be determined
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If the sum of 5 consecutive positive multiples of a prime number is 10  [#permalink]

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New post 11 Jun 2017, 03:42
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"If the sum of 5 consecutive positive multiples of a prime number is 105, what is the sum of the first 5 positive multiples of that prime number?"

Let P be the prime number.
In terms of equation the first part of the statement becomes: P*x + P*(x+1) + P*(x+2) + P*(x+3) + P*(x+4)=105
->P*(5x) + 10P=105
->P*(x+2)=21

There are 2 possibilities: (P,x)= (7,1) or (3,5).
(7,1) -> 7 + 14 + 21 + 28 + 35= 105
(3,5) -> 15 + 18 + 21 + 24 + 27= 105

If 7 is the prime number, the sum of the first 5 positive multiples would be 7 + 14 + 21 + 28 + 35= 105.
If 3 is the prime number, the sum of the first 5 positive multiples would be 3 + 6 + 9 + 12 + 15= 45.

--> Answer E
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Re: If the sum of 5 consecutive positive multiples of a prime number is 10  [#permalink]

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New post 01 Jun 2017, 14:34
Since the sum of 5 consecutive positive multiples of a prime number is 105, the mean is 21.
The numbers whose sum is 105 are 19,20,21,22,23.
Hence the prime number we are looking for is 19.
Now, we need the sum of the first 5 positive multiples of this prime number which is not possible
as the only positive multiples that a prime number has are the prime number and 1.

Since we need sum of first 5 positive multiples of this prime number it cannot be determined(Option E)
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Re: If the sum of 5 consecutive positive multiples of a prime number is 10  [#permalink]

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New post 02 Jun 2017, 23:31
How I did it- Please let me know if I my thought process is incorrect

Let the prime number be p
5 consecutive multiples for p would be pa, p(a+1), p(a+2),p(a+3), p(a+4)

Sum of 5 consecutive multiples of p=
=p[a+(a+1)+(a+2)+(a+3)+(a+4)]
=p[5/2(2a+4)] (Sum of AP)

105=p[5/2(2a+4)]

Thus we will not be able to determine p and a using only one equation .
Hence cannot be determined
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Re: If the sum of 5 consecutive positive multiples of a prime number is 10  [#permalink]

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New post 04 Jun 2017, 03:37
Hi,
Choose a prime number - 5
5 + 10+ 15 + 20 + 25 = 75
Ans could be c

15 + 20 + 25 + 30 + 35 > 105
10 + 15 + 20 + 25 + 30 > 105

Choose another prime number - 7
7 + 14+ 21 + 28 + 35 = 105 Ans D
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If the sum of 5 consecutive positive multiples of a prime number is 10  [#permalink]

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New post Updated on: 11 Jun 2017, 04:21
niteshwaghray wrote:
If the sum of 5 consecutive positive multiples of a prime number is 105, what is the sum of the first 5 positive multiples of that prime number?


A. 30
B. 45
C. 75
D. 105
E. Can't be determined



Hi,

Let the no. be P.
As per the question, \(P*x + P*(x+1) + P*(x+2) + P*(x+3) + P*(x+4) = 105\) --- eqn. 1
Solving the equation, we get :

\(5*P*x + 10*P = 105
=> P*x + 2*P = 21
=> P(x+2) = 21\)

Factors of 21 - 3,7,21

So\(P*(x+2)=21\) can either be \(7*(1+2)\) or\(3*(1+6).\)

Lets try the values : If x=1 and P=7, substituting in eqn. 1, we get \(7+14+21+28+35 = 105\)--- Satisfied
but if we consider P=3 and x=6, substituting in equation leads to terms - \(18+21+24+27+30 = 120\) which doesn't suffice.

Thus, P must be equal to 7 and x must be equal to 1.

Based on these, the sum of 1st 5 multiples = \(7+14+21+28+35 =105\)


As per me, option D should be the answer, can somebody please suggest why is it E ?

Got it. :oops: Such a careless mistake.

Originally posted by Shruti0805 on 11 Jun 2017, 01:39.
Last edited by Shruti0805 on 11 Jun 2017, 04:21, edited 1 time in total.
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Re: If the sum of 5 consecutive positive multiples of a prime number is 10  [#permalink]

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New post 11 Jun 2017, 03:32
Imo E
As we have two unknowns hence we can not determine sum of multiples .
Let the prime number be X and and Y be some positive integer .
Now according to question we have , X[y+(y+1)+(y+2)+(y+3)+(y+4)
= X[5y+10]
Thus we have two unknowns that can not determine answer



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Re: If the sum of 5 consecutive positive multiples of a prime number is 10  [#permalink]

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Re: If the sum of 5 consecutive positive multiples of a prime number is 10   [#permalink] 24 Feb 2019, 10:58
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