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GMAT/Data Sufficiency - Doesn't Add Up
Doesn't Add Up
a is the sum of x consecutive positive integers. b is the sum of y consecutive positive integers. For which of the following values of x and y is it impossible that a = b?
(A) x = 2; y = 6
(B) x = 3; y = 6
(C) x = 7; y = 9
(D) x = 10; y = 4
(E) x = 10; y = 7
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| Решение |
We can express the sum of consecutive integers algebraically. For example, the sum of three consecutive integers can be expressed as n + (n + 1) + (n + 2) = 3n + 3. So we need to see which values of x and y cannot be equated algebraically.
(A) Can the sum of two consecutive integers be equal to the sum of 6 consecutive integers?
We can express this as r + (r + 1) = s + (s + 1) + (s + 2) + (s + 3) + (s + 4) + (s + 5) --> 2r + 1 = 6s + 15.
Subtract 1 from both sides: 2r = 6s + 14.
Divide both sides by 2: r = 3s + 7. So if s = 2, for example, then r = 13.
Is it true that 13 + 14 = 2 + 3 + 4 + 5 + 6 + 7? Yes, 27 = 27. So this pair of values can work.
(B) We can express this as 3r + 3 = 6s + 15.
Divide through by 3: r + 1 = 2s + 5, subtract 1 from both sides: r = 2s + 4.
Whatever the value of s, we will find an integer value of r. This can work.
(C) We can express this as 7r + 21 = 9s + 36.
Subtract 21 from both sides: 7r = 9s + 15.
If s = 3, then 7r = 27 + 15 = 42, and r = 6. This can work.
(D) We can express this as 10r + 45 = 4s + 6.
We can see here that the left side will be odd (10r is even and 45 is odd --> even + odd = odd).
But the right side will be even (4s is even and 6 is even --> even + even = even).
Since an odd sum can never equal an even sum, these cannot be equal.
This cannot work and is therefore the correct answer.
(E) We can express this as 10r + 45 = 7s + 21.
Subtract 21 from both sides: 10r + 24 = 7s.
If r = 6, then 10(6) + 24 = 7s --> 60 + 24 = 7s --> 84 = 7s and s therefore is equal to 12. This can work.
The correct answer is D.
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Digit Doozey
What is the three-digit number abc, given that a, b, and c are the positive
single digits that make up the number?
(1) a = 1.5b and b = 1.5c
(2) a = 1.5x + b and b = x + c, where x represents a positive single digit
(A) Statement (1) alone is sufficient, but statement (2) alone is not
sufficient.
(B) Statement (2) alone is sufficient, but statement (1) alone is not
sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is
sufficient.
(D) Each statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient.
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Вверх 
