This Week's Challenge Problem from MGMAT : "Consecutive Positive Madness"https://www.manhattangmat.com/challenge_thisweek.cfm?submitted=1
Which of the following cannot be the sum of two or more consecutive positive integers?
(A) 3^7 (B) 4^6 (C) 5^5 (D) 6^4 (E) 7^3
I got this wrong ... Can anybody please explain?
There are lots of properties of numbers and GMAT does not expect you to know them. So a question that appears of GMAT must be solvable without knowing the properties. So think hard about what you know and what you can apply. Use pattern recognition.
Try some numbers to start off:
1+2 = 3
2+3 = 5
3+4 = 7
ok, so is there a pattern here? We are getting all odd numbers. 3, 5, 7, 9, 11 etc. Every odd number can be written as sum of two consecutive numbers. Why? Say an odd number is N. When you divide it by 2, you get half of it which has a .5. You take the integer above it and below it and they will add up to give N
e.g. N = 11.
11/2 = 5.5 so take numbers 5 and 6 and they will add to give 11. Why? because 5.5 is the arithmetic mean of 2 consecutive numbers:5 and 6.
Takeaway: Every odd number can be written as sum of two consecutive integers.
So rule out (A), (C) and (E).
Now, try to sum up three consecutive numbers.
1+2+3 = 6
2+3 +4 = 9 (ignore odd numbers so we have already dealt with them)
3+4+5 = 12
4+5+6 = 15
5+6+7 = 18
6+7+8 = 21
7+8+9 = 24
You are getting all multiples of 3. The important thing is that no multiple of 3 is getting skipped. You are getting all of them. Hence we can represent all multiples of 3 as sum of 3 numbers. Hence (D) is also out since it is a multiple of 3.
Now think why?
Sum of three consecutive integers is given by (n-1) + n + (n + 1) = 3n
Takeaway: Sum of any three consecutive numbers will be a multiple of 3.
Hence answer must be (B) i.e. 4^6
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