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12 Jul 2018, 01:11
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[GMAT math practice question]
When the positive integer n is divided by 8, the remainder is 3, and when n is divided by 5, the remainder is 2. What is the remainder when the smallest possible value of n is divided by 6?
A. 0
B. 1
C. 2
D. 3
E. 4
=>
If n has remainder 3 when it is divided by 8, then n = 8a + 3 for some integer a. The possible positive integer values of n are 3, 11, 19, 27, 35, … .
If n has remainder 2 when it is divided by 5, then n = 5b + 2 for some integer b. The possible positive integer values of n are 2, 7, 12, 17, 22, 27, 32, … .
The smallest value of n that appears in both lists is 27. Therefore, the smallest possible value for n is 27.
27 = 6*4 + 3, 27 has remainder 3 when it is divided by 6.
Therefore, the answer is D.
Answer: D
When the positive integer n is divided by 8, the remainder is 3, and when n is divided by 5, the remainder is 2. What is the remainder when the smallest possible value of n is divided by 6?
A. 0
B. 1
C. 2
D. 3
E. 4
=>
If n has remainder 3 when it is divided by 8, then n = 8a + 3 for some integer a. The possible positive integer values of n are 3, 11, 19, 27, 35, … .
If n has remainder 2 when it is divided by 5, then n = 5b + 2 for some integer b. The possible positive integer values of n are 2, 7, 12, 17, 22, 27, 32, … .
The smallest value of n that appears in both lists is 27. Therefore, the smallest possible value for n is 27.
27 = 6*4 + 3, 27 has remainder 3 when it is divided by 6.
Therefore, the answer is D.
Answer: D
15 Jul 2018, 07:02
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[GMAT math practice question]
The terms of the sequence {An} satisfy A2 - A1 = 2, A3 - A2 = 5 and A4 - A3 = 10. Which of the following could be the formula for An+1 - An, in terms of n?
A. n + 1
B. n^2 + 1
C. n^2 - 1
D. n - 1
E. n^2 + 3
=>
We try each possibility until we find a formula that works.
A): If An+1 - An = n + 1, then A2 - A1 = 1 + 1 = 2, which is correct, but
A3 - A2 = 2 + 1 = 3 ≠ 5.
So, A) is not the answer.
B): If An+1 - An = n^2 + 1, then
A2 - A1 = 1^2 + 1 = 2,
A3 - A2 = 2^2 + 1 = 5,
A4 - A3 = 3^2 + 1 = 10.
So, this formula is possible.
Therefore, the answer is B.
Answer: B
The terms of the sequence {An} satisfy A2 - A1 = 2, A3 - A2 = 5 and A4 - A3 = 10. Which of the following could be the formula for An+1 - An, in terms of n?
A. n + 1
B. n^2 + 1
C. n^2 - 1
D. n - 1
E. n^2 + 3
=>
We try each possibility until we find a formula that works.
A): If An+1 - An = n + 1, then A2 - A1 = 1 + 1 = 2, which is correct, but
A3 - A2 = 2 + 1 = 3 ≠ 5.
So, A) is not the answer.
B): If An+1 - An = n^2 + 1, then
A2 - A1 = 1^2 + 1 = 2,
A3 - A2 = 2^2 + 1 = 5,
A4 - A3 = 3^2 + 1 = 10.
So, this formula is possible.
Therefore, the answer is B.
Answer: B
15 Jul 2018, 18:20
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[GMAT math practice question]
If the sequence {An} satisfies An = An-1 - An-2, A1 = 0, and A2 = 1, where n is an integer greater than 2, then what is the sum of the first 100 terms of {An}?
A. 1
B. 2
C. 3
D. 4
E. 5
=>
We determine the period of the sequence by examining its terms:
A1 = 0 and A2 = 1.
A3 = A2 – A1 = 1 – 0 = 1
A4 = A3 – A2 = 1 – 1 = 0
A5 = A4 – A3 = 0 – 1 = -1
A6 = A5 – A4 = -1 – 0 = -1
A7 = A6 – A5 = -1 – (-1) = 0
A8 = A7 – A6 = 0 – (-1) = 1
A9 = A8 – A7 = 1 – 0 = 1
A10 = A9 – A8 = 1 – 1 = 0
…
The sequence has period 6.
The sum of the first six terms is A1 + A2 + … A6 = 0 + 1 + 1 + 0 + (-1) + (-1) = 0.
We apply this fact to determine the sum of the first 100 terms of the sequence:
A1 + A2 + … A100
= ( A1 + A2 + … A6 ) + … + ( A91 + A92 + … A96 ) + A97 + A98 + A99 + A100
= 0 + … + 0 + A97 + A98 + A99 + A100
= A97 + A98 + A99 + A100
= 0 + 1 + 1 + 0 = 2.
Therefore, the answer is B.
If the sequence {An} satisfies An = An-1 - An-2, A1 = 0, and A2 = 1, where n is an integer greater than 2, then what is the sum of the first 100 terms of {An}?
A. 1
B. 2
C. 3
D. 4
E. 5
=>
We determine the period of the sequence by examining its terms:
A1 = 0 and A2 = 1.
A3 = A2 – A1 = 1 – 0 = 1
A4 = A3 – A2 = 1 – 1 = 0
A5 = A4 – A3 = 0 – 1 = -1
A6 = A5 – A4 = -1 – 0 = -1
A7 = A6 – A5 = -1 – (-1) = 0
A8 = A7 – A6 = 0 – (-1) = 1
A9 = A8 – A7 = 1 – 0 = 1
A10 = A9 – A8 = 1 – 1 = 0
…
The sequence has period 6.
The sum of the first six terms is A1 + A2 + … A6 = 0 + 1 + 1 + 0 + (-1) + (-1) = 0.
We apply this fact to determine the sum of the first 100 terms of the sequence:
A1 + A2 + … A100
= ( A1 + A2 + … A6 ) + … + ( A91 + A92 + … A96 ) + A97 + A98 + A99 + A100
= 0 + … + 0 + A97 + A98 + A99 + A100
= A97 + A98 + A99 + A100
= 0 + 1 + 1 + 0 = 2.
Therefore, the answer is B.
