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# The terminal zeros of a number are the zeros to the right of its last

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Math Revolution GMAT Instructor
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The terminal zeros of a number are the zeros to the right of its last  [#permalink]

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09 Oct 2018, 19:34
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[Math Revolution GMAT math practice question]

The terminal zeros of a number are the zeros to the right of its last nonzero digit. For example, 30,500 has two terminal zeros because there are two zeros to the right of its last nonzero digit, 5. How many terminal zeros does n! have?

$$1) n^2 – 15n + 50 < 0$$
$$2) n > 5$$

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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" NUS School Moderator Joined: 18 Jul 2018 Posts: 1021 Location: India Concentration: Finance, Marketing WE: Engineering (Energy and Utilities) The terminal zeros of a number are the zeros to the right of its last [#permalink] ### Show Tags Updated on: 09 Oct 2018, 19:52 From statement 1: solving the quadratic equation gives (n-5)(n-10)<0. Or 5<n<10. Max value of n can be 9. And 9! has 1 terminating zero. A is sufficient. From statement 2: n>5. But n can take any value. If n is 9, then 9! Will have 1 zero. If n is 11, then 11! Will have 2 zeros. Hence B is insufficient. A is the answer Posted from my mobile device _________________ Press +1 Kudos If my post helps! Originally posted by Chethan92 on 09 Oct 2018, 19:43. Last edited by Chethan92 on 09 Oct 2018, 19:52, edited 1 time in total. Math Expert Joined: 02 Aug 2009 Posts: 8004 Re: The terminal zeros of a number are the zeros to the right of its last [#permalink] ### Show Tags 09 Oct 2018, 19:44 The terminal zeros of a number are the zeros to the right of its last nonzero digit. For example, 30,500 has two terminal zeros because there are two zeros to the right of its last nonzero digit, 5. How many terminal zeros does n! have? $$1) n^2 – 15n + 50 < 0$$ $$n^2-10n-5n+50<0......(n-10)(n-5)<0$$ So n is between 5 and 10 For each of 6!,7!,8! or 9!, Number of zeroes will be 1 as only 1*5 is there in each Sufficient $$2) n > 5$$ If 6 ANS is 1 and if 10, Ans is 2 Insufficient A _________________ Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8025 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: The terminal zeros of a number are the zeros to the right of its last [#permalink] ### Show Tags 11 Oct 2018, 01:34 => Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution. Since we have 1 variable (n) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first. Consider 1) $$n^2 – 15n + 60 < 0$$ $$=> (n-5)(n-10) < 0$$ $$=> 5 < n < 10$$ The number of terminal zeros of a number is determined by the number of $$5s$$ in its prime factorization. The integers satisfying $$5 < n < 10$$ are $$6, 7, 8$$ and $$9$$. We count the $$5s$$ in the prime factorizations of $$6!, 7!, 8!$$ and $$9!$$: $$6!$$ has one 5 in its prime factorization. $$7!$$ has one 5 in its prime factorization. $$8!$$ has one 5 in its prime factorization. $$9!$$ has one 5 in its prime factorization. Thus, for $$5 < n < 10, n!$$ has one terminal zero. As it gives us a unique answer, condition 1) is sufficient. Condition 2) If $$n = 6$$, then $$6 > 5$$ and $$6! = 720$$ has one terminal $$0$$. If $$n = 10$$, then $$10 > 5$$ and $$10! = 3,628,800$$ has two terminal $$0s$$. Condition 2) is not sufficient since it does not give a unique solution. Therefore, A is the answer. Answer: A If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E. _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
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Re: The terminal zeros of a number are the zeros to the right of its last  [#permalink]

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11 Oct 2018, 07:37
MathRevolution wrote:
[Math Revolution GMAT math practice question]

The terminal zeros of a number are the zeros to the right of its last nonzero digit. For example, 30,500 has two terminal zeros because there are two zeros to the right of its last nonzero digit, 5. How many terminal zeros does n! have?

$$1) n^2 – 15n + 50 < 0$$
$$2) n > 5$$

$$?\,\,\,:\,\,\,{\text{number}}\,\,{\text{of}}\,\,{\text{terminal}}\,\,{\text{zeros}}\,\,{\text{of}}\,\,n\,{\text{!}}$$

$$\left( 1 \right)\,\,\,{n^2} - 15n + 50 < 0\,\,\,\, \Leftrightarrow \,\,\,\,5 < n < 10$$

$$\left. \begin{gathered} n! = 6! = 6 \cdot \boxed5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\,\,\,\,\, \Rightarrow \,\,\,\,\,\frac{{6!}}{{10}} = \operatorname{int} \,\,\,\,\,{\text{but}}\,\,\,\,\,\frac{{6!}}{{{{10}^2}}} \ne \operatorname{int} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 1\,\, \hfill \\ n! = 7! = 7 \cdot 6 \cdot \boxed5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\,\,\,\,\, \Rightarrow \,\,\,\,\,\frac{{7!}}{{10}} = \operatorname{int} \,\,\,\,\,{\text{but}}\,\,\,\,\,\frac{{7!}}{{{{10}^2}}} \ne \operatorname{int} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 1\,\, \hfill \\ n! = 8! = 8 \cdot 7 \cdot 6 \cdot \boxed5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\,\,\,\,\, \Rightarrow \,\,\,\,\,\frac{{8!}}{{10}} = \operatorname{int} \,\,\,\,\,{\text{but}}\,\,\,\,\,\frac{{8!}}{{{{10}^2}}} \ne \operatorname{int} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 1\, \hfill \\ n! = 9! = 9 \cdot 8 \cdot 7 \cdot 6 \cdot \boxed5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\,\,\,\,\, \Rightarrow \,\,\,\,\,\frac{{9!}}{{10}} = \operatorname{int} \,\,\,\,\,{\text{but}}\,\,\,\,\,\frac{{9!}}{{{{10}^2}}} \ne \operatorname{int} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 1\,\, \hfill \\ \end{gathered} \right\}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 1$$

$$\left( 2 \right)\,\,n > 5\,\,\,\,\,\left\{ \begin{gathered} \,{\text{Take}}\,\,n = 6\,\,\,\, \Rightarrow \,\,\,? = 1\,\, \hfill \\ \,{\text{Take}}\,\,n = 10\,\, \Rightarrow \,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{2}}\,\, \hfill \\ \end{gathered} \right.$$

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

Regards,
Fabio.
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Re: The terminal zeros of a number are the zeros to the right of its last  [#permalink]

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11 Oct 2018, 20:03
chetan2u wrote:
The terminal zeros of a number are the zeros to the right of its last nonzero digit. For example, 30,500 has two terminal zeros because there are two zeros to the right of its last nonzero digit, 5. How many terminal zeros does n! have?

$$1) n^2 – 15n + 50 < 0$$
$$n^2-10n-5n+50<0......(n-10)(n-5)<0$$
So n is between 5 and 10
For each of 6!,7!,8! or 9!, Number of zeroes will be 1 as only 1*5 is there in each
Sufficient

$$2) n > 5$$
If 6 ANS is 1 and if 10, Ans is 2
Insufficient

A

n2−10n−5n+50<0......(n−10)(n−5)<0
How did we solve this to get 5<n<10. Wont it be n<5 and n<10.Please help me solve these type of inequality.
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Re: The terminal zeros of a number are the zeros to the right of its last  [#permalink]

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12 Oct 2018, 01:13
chetan2u wrote:
The terminal zeros of a number are the zeros to the right of its last nonzero digit. For example, 30,500 has two terminal zeros because there are two zeros to the right of its last nonzero digit, 5. How many terminal zeros does n! have?

$$1) n^2 – 15n + 50 < 0$$
$$n^2-10n-5n+50<0......(n-10)(n-5)<0$$
So n is between 5 and 10
For each of 6!,7!,8! or 9!, Number of zeroes will be 1 as only 1*5 is there in each
Sufficient

$$2) n > 5$$
If 6 ANS is 1 and if 10, Ans is 2
Insufficient

A

n2−10n−5n+50<0......(n−10)(n−5)<0
How did we solve this to get 5<n<10. Wont it be n<5 and n<10.Please help me solve these type of inequality.

Have the same doubt. Experts,kindly enlighten. Thank you. Bunuel VeritasKarishma
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Joined: 18 Jul 2018
Posts: 1021
Location: India
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The terminal zeros of a number are the zeros to the right of its last  [#permalink]

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12 Oct 2018, 01:19
chetan2u wrote:
The terminal zeros of a number are the zeros to the right of its last nonzero digit. For example, 30,500 has two terminal zeros because there are two zeros to the right of its last nonzero digit, 5. How many terminal zeros does n! have?

$$1) n^2 – 15n + 50 < 0$$
$$n^2-10n-5n+50<0......(n-10)(n-5)<0$$
So n is between 5 and 10
For each of 6!,7!,8! or 9!, Number of zeroes will be 1 as only 1*5 is there in each
Sufficient

$$2) n > 5$$
If 6 ANS is 1 and if 10, Ans is 2
Insufficient

A

n2−10n−5n+50<0......(n−10)(n−5)<0
How did we solve this to get 5<n<10. Wont it be n<5 and n<10.Please help me solve these type of inequality.

The highlighted portion is an inequality. solving gives n = 5 and 10. plotting this on the number line using wavy curve method. Refer to the attached file.

Since the inequality sign is <0. we should consider the integers in the range 5<n<10.

Refer this topic https://gmatclub.com/forum/wavy-line-method-application-complex-algebraic-inequalities-224319.html for wavy curve method
Attachments

Untitled.png [ 4.41 KiB | Viewed 318 times ]

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Re: The terminal zeros of a number are the zeros to the right of its last  [#permalink]

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12 Oct 2018, 01:33
Afc0892 wrote:
chetan2u wrote:
The terminal zeros of a number are the zeros to the right of its last nonzero digit. For example, 30,500 has two terminal zeros because there are two zeros to the right of its last nonzero digit, 5. How many terminal zeros does n! have?

$$1) n^2 – 15n + 50 < 0$$
$$n^2-10n-5n+50<0......(n-10)(n-5)<0$$
So n is between 5 and 10
For each of 6!,7!,8! or 9!, Number of zeroes will be 1 as only 1*5 is there in each
Sufficient

$$2) n > 5$$
If 6 ANS is 1 and if 10, Ans is 2
Insufficient

A

n2−10n−5n+50<0......(n−10)(n−5)<0
How did we solve this to get 5<n<10. Wont it be n<5 and n<10.Please help me solve these type of inequality.

The highlighted portion is an inequality. solving gives n = 5 and 10. plotting this on the number line using wavy curve method. Refer to the attached file.

Since the inequality sign is <0. we should consider the integers in the range 5<n<10.

Refer this topic https://gmatclub.com/forum/wavy-line-method-application-complex-algebraic-inequalities-224319.html for wavy curve method

Afc0892 thank you for the explanation!
Re: The terminal zeros of a number are the zeros to the right of its last   [#permalink] 12 Oct 2018, 01:33
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# The terminal zeros of a number are the zeros to the right of its last

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