If t is a positive integer, can (t+2)(t-3) be evenly divided by 6?(1) 5(t3 +1) is not divisible by 2.(2) t is a 3-digit number, whose digits are consecutive integers.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.Correct answer is option 'B'. Can you explain this answer?

Question

Answer

Solution:

We need to find out whether (t^2)(t-3) is divisible by 6 or not.

Statement 1:

5(t^3+1) is not divisible by 2.

This tells us that t is an odd number. However, we cannot determine whether (t^2)(t-3) is divisible by 6 or not based on this information alone.

Statement 2:

t is a 3-digit number whose digits are consecutive integers.

We can write t as (t-1)(t)(t+1). Since t is a 3-digit consecutive integer, it must be divisible by 3. Therefore, either t or (t-3) must be divisible by 3. Also, one of t-1 and t+1 must be divisible by 2. Hence, we can conclude that (t-3)t(t-1) is divisible by 6.

Combining Statements 1 and 2:

From statement 1, we know that t is an odd number. From statement 2, we know that (t-3)t(t-1) is divisible by 6. However, since t is odd, either (t-3) or (t-1) must be even, which means that (t-3)t(t-1) is divisible by 2. Therefore, (t-3)t(t-1) is divisible by both 2 and 6, which means it is divisible by 6.

Conclusion:

Statement 2 alone is sufficient to answer the question, while statement 1 alone is not. Therefore, the correct answer is option B.

Answer

Let's evaluate each statement separately:

Statement (1): 5(t^3 + 1) is not divisible by 2. Thisstatement tells us that 5(t^3 + 1) is not divisible by 2. Since t is a positiveinteger, t^3 + 1 will always be an odd number. Odd numbers multiplied by 5 willremain odd. We can conclude that t^3 + 1 is not divisible by 2. However, thisstatement does not provide any information about the divisibility of (t + 2)(t- 3) by 2 or 3.

Statement (2): t is a 3-digit number, whose digits areconsecutive integers. This statement provides information about the nature oft. If the digits of t are consecutive integers, we can write t as t = x(x +1)(x + 2), where x is an integer. Multiplying this expression out, we get t =x^3 + 3x^2 + 2x. Now let's substitute this expression for t into (t + 2)(t - 3)and check its divisibility by 6:

(t + 2)(t - 3) = (x^3 + 3x^2 + 2x + 2)(x^3 + 3x^2 + 2x - 3)Expanding the expression: = x^6 + 6x^5 + 13x^4 + 10x^3 - 4x^2 - 6x - 6 We canobserve that this expression contains terms with x^6 and x^5, which are bothdivisible by 6. Therefore, (t + 2)(t - 3) is divisible by 6 based on the givenconditions in statement (2).

By analyzing each statement, we find that statement (2)alone is sufficient to determine that (t + 2)(t - 3) is divisible by 6. Thus,the correct answer is option 'B': Statement (2) ALONE is sufficient, butstatement (1) alone is not sufficient.

Similar GMAT Doubts

If n is an integer, then n is divisible by how many positive integers?(1) n is the product of two different prime numbers.(2) n and 23 are each divisible by the same number of positive integers.

If x, y, z are positive integers, is xyz divisible by 6?(1) x, y, z are consecutive.(2) x + y + z is a multiple of 3.

The function SCI is defined as SCI(x, y) = z, where z is the sum of y consecutive positive integers starting from positive integer x. If a and n are positive integers, is SCI(a,n) divisible by n?(1) 3n +2a is not divisible by 2(2) 3a +2n is divisible by 2

If a, b, c and d are positive consecutive multiples, not necessarily in that order, of a positive integer x greater than 1, is a + b + c + d 50?(1) c = 15(2) The difference between d and b is divisible by only four positive integers, one of which is 10.

The function SCI is defined as SCI(x, y) = z, where z is the sum of y consecutive positive integers starting from positive integer x. If a and n are positive integers, is SCI(a,n) divisible by n?(1) 3n +2a is not divisible by 2(2) 3a +2n is divisible by 2

Top Courses for GMATView all

GMAT Focus Edition Mock Test Series 2024

Quantitative for GMAT

Crash Course for GMAT

Data Insights for GMAT

100 RCs for GMAT

Top Courses for GMAT

Top Courses for GMAT

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams