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Integers 3n+2 and 8n+3 are divisible by an integer p. If p is not equal to 1, then p equals to?
A. 2
B. 8
C. 7
D. 11
E. 6
A. 2
B. 8
C. 7
D. 11
E. 6
SergeNew
Trial and error approach would probably be the fastest for this problem.
One can also do: 8n+3=even+odd=odd, which means that p must be odd, so the answer is either 7 or 11. Now, since both numbers are divisible by p then their sum must also be divisible by p: (3n+2)+(8n+3)=11n+5. 11n+5 is not a multiple of 11 (it's 5 more than multiple of 11), hence p can only be 7.
Answer: C.
Hope it's clear.
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Not sure if there is a shortest way to do this.
I just substituted values from 1 to 4 for n in both the equations 3n+2 and 8n+3.
When n=1, 3n+2= 5, 8n+3= 11 - No common divisor
When n=2, 3n+2= 8, 8n+3= 19 - No common divisor
When n=3, 3n+2= 11, 8n+3= 27 - No common divisor
When n=4, 3n+2=14 , 8n+3= 35 - divisible by 7
Answer C
I just substituted values from 1 to 4 for n in both the equations 3n+2 and 8n+3.
When n=1, 3n+2= 5, 8n+3= 11 - No common divisor
When n=2, 3n+2= 8, 8n+3= 19 - No common divisor
When n=3, 3n+2= 11, 8n+3= 27 - No common divisor
When n=4, 3n+2=14 , 8n+3= 35 - divisible by 7
Answer C
General Discussion
subhashghosh
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I would also do the same, however, please note that 8n+3 is odd (because 8n is even and odd + even = odd). So the task is reduced to picking out from 7 and 11 among the answer choices now.
I used a more algebraic approach
From the question we can gather that
3n + 2 = pq and 8n + 3 = px
If we re-arrange in terms of n we get
n= (pq-2) / 3 and n= (px-3) / 8
equating the two equations we get
8(pq-2) = 3(px-3)
which simplifies to
p= 7 / (8q-3x)
therefore p is a multiple of 7
- answer C
Hope this helps
From the question we can gather that
3n + 2 = pq and 8n + 3 = px
If we re-arrange in terms of n we get
n= (pq-2) / 3 and n= (px-3) / 8
equating the two equations we get
8(pq-2) = 3(px-3)
which simplifies to
p= 7 / (8q-3x)
therefore p is a multiple of 7
- answer CHope this helps
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SergeNew
3n + 2 is not divisible by 3 so 6 is ruled out.
8n + 3 is odd and not divisible by both 3 and 8 .So only 7 and 11 are left out.
I used trial and error after this.
SergeNew
let x and y be quotients
3n+2=px→24n+16=8px
8n+3=py→24n+9=3py
subtracting,
7=8px-3py→7/p=8x-3y
if 8x-3y is an integer, then
p=7
C
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SergeNew
Let’s list the pairs (3n + 2, 8n + 3) for some values of n, starting from n = 0:
(2, 3), (5, 11), (8, 19), (11, 27), (14, 35), …
We see that the first four pairs are relatively prime (i.e. they have no common factors besides 1), but the fifth pair (14, 35) has the common factor of 7. Thus, p is 7.
Answer: C
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Posts: 1
Integers 3n+2 and 8n+3 are divisible by an integer p
then (8n+3) - (3n+2) = 5n+1 also divisible by integer p
now we have 3n+2, 5n+1, 8n+3
then do the calculation again: (5n+1) - (3n+2) = 2n-1
so we have 2n-1, 3n+2, 5n+1, 8n+3 all divisible by p
then do the calculation again: (3n+2) - (2n-1) = n+3
so we get n+3, 2n-1, 3n+2, 5n+1, 8n+3 all divisible by p
since n+3 divisible by p, we can imply (n+3) + (n+3) = 2n+6 divisible by p
so (2n+6) - (2n-1) = 7 and we get the answer.
then (8n+3) - (3n+2) = 5n+1 also divisible by integer p
now we have 3n+2, 5n+1, 8n+3
then do the calculation again: (5n+1) - (3n+2) = 2n-1
so we have 2n-1, 3n+2, 5n+1, 8n+3 all divisible by p
then do the calculation again: (3n+2) - (2n-1) = n+3
so we get n+3, 2n-1, 3n+2, 5n+1, 8n+3 all divisible by p
since n+3 divisible by p, we can imply (n+3) + (n+3) = 2n+6 divisible by p
so (2n+6) - (2n-1) = 7 and we get the answer.
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Since I know if I subtract the two terms the result will be divible by p
So 3n + 2 -8n -3
-5n-1
Second stage
-5n -1 -8n -3
-13n -4
Third stage
-13n - 4 - 8n - 3
-21n - 7
Now that is divisible by 7
Answer choice C
So 3n + 2 -8n -3
-5n-1
Second stage
-5n -1 -8n -3
-13n -4
Third stage
-13n - 4 - 8n - 3
-21n - 7
Now that is divisible by 7
Answer choice C
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SergeNew
If \(n = 4\) , \(3n+2 = 14\) & \(8n+3 = 35\)
\(HCF (14 , 35) = 7\), Answer must be (C)
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Integers 3n+2 and 8n+3 are divisible by an integer p. Therefore 8*(3n+2) and 3(8n+3) are divisible by 3. Which are 24n+16 and 24n+9 .The difference is 7 which should also be multiple of p. since the option has only 7 which divides .Answer is 7 (option c)
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