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The sum of the positive integers from 1 through n can be calculated by the formula n(n + 1)/2. Which one of the following equals the sum of all the even numbers from 0 through 20, inclusive? (A) (B) (C) (D) (E) 50 70 90 110 140 26. a, b, c, d, and e are five consecutive numbers in increasing order of size. Deleting one of the five numbers from the set decreased the sum of the remaining numbers in the set by 20%. Which one of the following numbers was deleted? (A) (B) (C) (D) (E) a b c d e 27. A set has exactly five consecutive positive integers starting with 1.
The answer is (D). 1 bh, where b is the base and h is the height. Sometimes the base must be extended 2 in order to draw the altitude, as in the third drawing directly below: The area of a triangle is h h A= h b b 1 bh 2 b In a triangle, the longer side is opposite the larger angle, and vice versa: a 100˚ b 50˚ 50˚ is larger than 30˚, so side b is longer than side a. 30˚ c Pythagorean Theorem (right triangles only): The square of the hypotenuse is equal to the sum of the squares of the legs. c a c 2 = a2 + b 2 b Pythagorean triples: The numbers 3, 4, and 5 can always represent the sides of a right triangle and they appear very often: 52 = 32 + 4 2 .
Then by the Difference of Squares formula a2 – b 2 = (a – b)(a + b) = 4(a + b) [Given that difference between the digits is 4]. This is maximum when (a + b) is maximum. If a and b are single digit numbers, then a + b would be a maximum when a is equal to 9 and b = 9 – 4 = 5 (given the numbers differ by 4). Hence, the maximum value of a 2 – b2 is 9 2 – 52 = 56. The answer is (D). 43 44 GMAT Math Bible Very Hard 31. We are given that the numbers m and n, when divided by 6, leave remainders of 2 and 3, respectively.