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Challenge question: j and k are positive integers

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Challenge question: j and k are positive integers

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j and k are positive integers, and n = 10^j + k. Is n divisible by 15?
(1) j and k are each divisible by 3
(2) j and k are each divisible by 5

Answer: A
Difficulty level: 700+
Source: www.gmatprepnow.com

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Brent@GMATPrepNow wrote:
j and k are positive integers, and n = 10^j + k. Is n divisible by 15?
(1) j and k are each divisible by 3
(2) j and k are each divisible by 5

Source: www.gmatprepnow.com
$$n = {\rm{1}}{{\rm{0}}^{\rm{j}}} + k\,\,\,\,;\,\,\,\,\,j,k\,\,\, \ge 1\,\,\,\,{\rm{ints}}\,\,\,\left( * \right)$$
$${n \over {3 \cdot 5}}\,\,\,\mathop = \limits^? \,\,\,{\mathop{\rm int}} $$
$$\left( 1 \right)\,\, \cap \,\,\,\left( * \right)\,\,\, \Rightarrow \,\,\,\left\{ \matrix{
j = 3M,\,\,M \ge 1\,\,{\mathop{\rm int}} \hfill \cr
k = 3L,\,\,L \ge 1\,\,{\mathop{\rm int}} \hfill \cr} \right.$$
$$n\,\,\, = \underbrace {{{\left( {{{10}^{M\, \ge \,1}}} \right)}^3}}_{\left\langle {100 \ldots 0} \right\rangle \,\,{\rm{not}}\,\,{\rm{div}}\,\,{\rm{by}}\,\,3} + \underbrace {3L}_{{\rm{div}}\,\,{\rm{by}}\,\,3}\,\, = \,\,{\rm{not}}\,\,{\rm{div}}\,\,{\rm{by}}\,\,\,3\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle $$
$$\left( 2 \right)\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {j,k} \right) = \left( {5,5} \right)\,\,\,\,\,\, \Rightarrow \,\,\,n = 100005\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\,\,\left( {\sum\nolimits_{{\rm{digits}}} {\,\,{\rm{and}}\,\,{\rm{final}}\,\,{\rm{digit}}} } \right) \hfill \cr
\,{\rm{Take}}\,\,\left( {j,k} \right) = \left( {5,10} \right)\,\,\,\,\,\, \Rightarrow \,\,\,n = 100010\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\left( {\sum\nolimits_{{\rm{digits}}} {} } \right) \hfill \cr} \right.$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

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Brent@GMATPrepNow wrote:
j and k are positive integers, and n = 10^j + k. Is n divisible by 15?
(1) j and k are each divisible by 3
(2) j and k are each divisible by 5
Given: n = 10^j + k

Target question: Is n divisible by 15?

Key concepts:
- If n is divisible by 15, then n must be divisible by 3 AND by 5
- If n is divisible by 3, then the sum of n's digits must be divisible by 3 (for example, we know that 747 is divisible by 3, because 7+4+7 = 18, and 18 is divisible by 3)


Statement 1: j and k are each divisible by 3
First recognize that 10^j = 1 followed by j zeros.
For example, 10^15 = 1,000,000,000,000,000 (1 followed by 15 zeros)
Next, recognize that, if k is divisible by 3, then the sum of n's digits must be divisible by 3
So, if n = 10^j + k, then the sum of n's digits will be 1 greater than some multiple of 3. [since we're adding 1 and several zeros to a number that is divisible by 3]

For example, if j = 6 and k = 24, then n = 10^j + k = 10^6 + 24 = 1,000,024. In this case, the sum of n's digits = 1+0+0+0+0+2+4 = 7, which is 1 greater than a multiple of 3.
Likewise, if j = 9 and k = 75, then n = 10^j + k = 10^9 + 75 = 1,000, 000,075. In this case, the sum of n's digits = 13, which is 1 greater than a multiple of 3.
And, if j = 15 and k = 99, then n = 10^15 + 99 = 1,000, 000,000,000,099. In this case, the sum of n's digits = 19, which is 1 greater than a multiple of 3.
And so on.
Since the sum of n's digits will always be 1 greater than some multiple of 3, we can be certain that n is NOT divisible by 3
So, by the above property, n is NOT divisible by 15
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: j and k are each divisible by 5
There are several values of j and k that satisfy statement 2. Here are two:
Case a: j = 5 and k = 5. In this case, n = 10^j + k = 10^5 + 5 = 100,005. Since 100,005 is divisible by 3 and by 5, the answer to the target question is YES, n IS divisible by 15
Case b: j = 5 and k = 10. In this case, n = 10^j + k = 10^5 + 10 = 100,010. Since 100,010 is NOT divisible by 3, the answer to the target question is NO, n is NOT divisible by 15
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

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Brent Hanneson – Creator of GMATPrepNow.com
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