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26 Jan 2021, 03:30
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
22% (02:04) correct
78%
(02:21)
wrong
based on 69
sessions
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If s is the sum of two prime numbers and p is the product of these prime numbers, which of the following could not be the value of p-s?
(A) 35
(B) 119
(C) 161
(D) 351
(E) 397
Are You Up For the Challenge: 700 Level Questions: 700 Level Questions
(A) 35
(B) 119
(C) 161
(D) 351
(E) 397
Are You Up For the Challenge: 700 Level Questions: 700 Level Questions
26 Jan 2021, 04:12
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Bunuel
If I have to make an intelligent guess, I will look into the property of the options.
We are talking of prime numbers and all except E are non-prime. I would mark E as the answer and move to the next question.
Let the two prime numbers be a and b.
so s=a+b, and p=ab
\(p-s=ab-(a+b)=ab-a-b=a(b-1)-b\)
To get it into more friendly terms, let us add -1+1 to it
\(a(b-1)-b+1-1=a(b-1)-1(b-1)-1=(a-1)(b-1)-1\)
Let us check the options which can be converted into the same form
(A) 35..........(a-1)(b-1)-1=35....(a-1)(b-1)=36=6*6, so a and b can be 7 each. Possible
(B) 119..........(a-1)(b-1)-1=119....(a-1)(b-1)=120=2*2*2*3*5=12*10, so a and b can be 13 and 11. Possible
(C) 161..........(a-1)(b-1)-1=161....(a-1)(b-1)=162=2*3*3*3*3=1*162, so a and b can be 2 and 163. Possible
(D) 351..........(a-1)(b-1)-1=351....(a-1)(b-1)=352=2*2*2*2*2*11=16*22, so a and b can be 17 and 23. Possible
(E) 397..........(a-1)(b-1)-1=397....(a-1)(b-1)=398=2*199=1*398, so a and b can be 3 and 200, but 200 is not a prime or 2*399, but 399 is not a prime. Not possible
Also what all does (a-1)(b-1)-1 tell you?
1) If we add one to the option, the answer is (a-1)(b-1).
So, when both a and b are odd prime numbers, a-1 and b-1 will be even, and we should get a multiple of 4. => 35+1, 119+1, 351+1 are all multiple of 4.
2) If a is even, that is it is 2, the other is odd, then (a-1)(b-1)=(2-1)(b-1)=b-1. That is when we add 1 to option+1, we should get a prime number. 161+2=163, which is prime. However, 397+2=399, which is not a prime.
E
03 Mar 2026, 01:03
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If s is the sum of two prime numbers and p is the product of these prime numbers, which of the following could not be the value of p-s?
Both p1 & p2 are odd prime numbers
s = p1 + p2
p = p1*p2
Let k = p - s = p1*p2 - (p1+p2)
k + 1 = p1*p2 - p1 - p2 + 1 = (p1-1)(p2-1)
(A) 35: k+1 = 36 = (7-1)(7-1) = (2-1)(37-1): Feasible
(B) 119: k+1 = 120 = (5-1)(31-1): Feasible
(C) 161: k+1 = 162 = (2-1)(163-1): Feasible
(D) 351: k+1 = 352 = (17-1)(23-1): Feasible
(E) 397
IMO E
Both p1 & p2 are odd prime numbers
s = p1 + p2
p = p1*p2
Let k = p - s = p1*p2 - (p1+p2)
k + 1 = p1*p2 - p1 - p2 + 1 = (p1-1)(p2-1)
(A) 35: k+1 = 36 = (7-1)(7-1) = (2-1)(37-1): Feasible
(B) 119: k+1 = 120 = (5-1)(31-1): Feasible
(C) 161: k+1 = 162 = (2-1)(163-1): Feasible
(D) 351: k+1 = 352 = (17-1)(23-1): Feasible
(E) 397
IMO E
