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# Mr. McCall selects a number that is two-digit and positive. If the num

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Math Expert
Joined: 02 Sep 2009
Posts: 55272
Mr. McCall selects a number that is two-digit and positive. If the num  [#permalink]

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07 Jul 2018, 09:42
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55% (hard)

Question Stats:

62% (01:55) correct 38% (01:43) wrong based on 175 sessions

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Mr. McCall selects a number that is two-digit and positive. If the number is prime, he assigns that many problems for homework. If the number is not prime, he assigns 8 more problems than the number for homework. If he assigns 97 problems for homework, which of the following could be the number he selected?

I. 89
II. 97
III. 105

A. I only
B. I and II only
C. II only
D. II and III only
E. I and III only

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Posts: 7686
Re: Mr. McCall selects a number that is two-digit and positive. If the num  [#permalink]

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07 Jul 2018, 10:28
Bunuel wrote:
Mr. McCall selects a number that is two-digit and positive. If the number is prime, he assigns that many problems for homework. If the number is not prime, he assigns 8 more problems than the number for homework. If he assigns 97 problems for homework, which of the following could be the number he selected?

I. 89
II. 97
III. 105

A. I only
B. I and II only
C. II only
D. II and III only
E. I and III only

check if 97 is a PRIME number
TIP :- check all prime numbers till $$\sqrt{97}~\sqrt{100}$$~10..

97 is odd, so not div by 2
9+7=16, so not div by 3
units digit is not 5, so not div by 5
97+70+27... 27 is not div by 7..so not div by 7
hence 97 is prime
so TWO cases
1) number picked up was 97 itself
2) number picked up was 97-8=89, but then 89 should not be a prime

check for 89 in similar manner
it is also prime, so case (2) is out

Only 97 possible

C
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Re: Mr. McCall selects a number that is two-digit and positive. If the num  [#permalink]

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07 Jul 2018, 11:13
Bunuel wrote:
Mr. McCall selects a number that is two-digit and positive. If the number is prime, he assigns that many problems for homework. If the number is not prime, he assigns 8 more problems than the number for homework. If he assigns 97 problems for homework, which of the following could be the number he selected?

I. 89
II. 97
III. 105

A. I only
B. I and II only
C. II only
D. II and III only
E. I and III only

No of problems assigned=97

Among 89,97, and 105; 89 and 97 are the prime numbers. So, 105 is eliminated.
89 can't be incremented to 97 since it's a prime; as per question algorithm stops at 89.

Hence,97 is the selected number.

Ans (C)
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Re: Mr. McCall selects a number that is two-digit and positive. If the num  [#permalink]

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07 Jul 2018, 12:17
1
Any prime number p, (p >= 5), when divided by 6 leaves a remainder of 1 or 5.

Since the home work given is of 97 problems, 97 divided by 6, leaves remainder of 1. Hence 97 is a prime number.

As per logic in question, if number chosen is prime, homework is equal to number chosen. Hence number chosen was 97.

Option II is true.

Since 89 is also prime number ( as it leaves a remainder of 5 on division by 6) & 97 = 89 + 8, therefore 89 is not possible.

Similarly for 105, which is not a prime number & 105 - 8 = 97, therefore 105 is also not possible.

Thanks,
GyM
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Re: Mr. McCall selects a number that is two-digit and positive. If the num  [#permalink]

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04 Aug 2018, 05:25
Bunuel wrote:
Mr. McCall selects a number that is two-digit and positive. If the number is prime, he assigns that many problems for homework. If the number is not prime, he assigns 8 more problems than the number for homework. If he assigns 97 problems for homework, which of the following could be the number he selected?

I. 89
II. 97
III. 105

A. I only
B. I and II only
C. II only
D. II and III only
E. I and III only

i think the wording of this question could be adjusted

If he assigns 97 problems for homework, which of the following could must be the number he selected?
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Re: Mr. McCall selects a number that is two-digit and positive. If the num  [#permalink]

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05 Dec 2018, 15:54
Hi All,

We're told that Mr. McCall selects a number that is two-digit and positive: If the number is prime, then he assigns that many problems for homework. If the number is NOT prime, then he assigns 8 MORE problems than the number for homework. We're told he assigns 97 problems for homework. We're asked which of the following could be the original number he selected. This question can be solved in a couple of different ways, including by TESTing THE ANSWERS.

To start, it's worth noting that there are only TWO potential ways to get to a specific number of questions, so only 1 or 2 of the three given values will 'fit' what we're told.

I. 89
The number 89 is prime. You can prove it by doing a little division. Since 89 is ODD, no even numbers will divide evenly into it and since it ends in '9', we know that 5 does not divide evenly in. Thus, you really just have to check 3, 7 and 9. The sum of the digits 8+9 = 17, so using the 'rule of 3' and the 'rule of 9', you can quickly eliminate those options. That just leaves the 7 - and 7 does NOT divide evenly into 89. Thus, Mr. McCall would have assigned 89 questions if he had chosen 89; he assigned 97 questions though, so this number is NOT correct.
Eliminate Answers A, B and E.

II. 97
Since Roman Numeral II is in both of the remaining answers, we KNOW that it's a valid option. To be thorough though, the number 97 is prime - and you can prove it in the exact same way that we proved that 89 is prime (see above).

III. 105
The number 105 is NOT prime; it actually has a number of different factors (the most obvious of which is 5, since 105 ends in a '5'). Thus, Mr. McCall would have assigned 8 MORE - meaning 105+8 = 113 questions - in this situation. This does NOT match the 97 questions that we're told he assigns though, so this number is NOT correct.

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Re: Mr. McCall selects a number that is two-digit and positive. If the num  [#permalink]

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10 Apr 2019, 17:53
Bunuel wrote:
Mr. McCall selects a number that is two-digit and positive. If the number is prime, he assigns that many problems for homework. If the number is not prime, he assigns 8 more problems than the number for homework. If he assigns 97 problems for homework, which of the following could be the number he selected?

I. 89
II. 97
III. 105

A. I only
B. I and II only
C. II only
D. II and III only
E. I and III only

Since 97 is a prime, he could have selected 97 and thus assigned 97 problems for homework.

He could also have selected a composite number (i.e., number that is not a prime) and assigned 97 problems for homework. To do so, let’s let x be the composite number he has selected, and thus we have:

x + 8 = 97

x = 89

However, 89 is not composite (89 is a prime), so he couldn’t have selected the number 89. Lastly, he couldn’t have selected 105, since 105 is not a two-digit number.

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Re: Mr. McCall selects a number that is two-digit and positive. If the num   [#permalink] 10 Apr 2019, 17:53
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