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When the positive integer n is divided by 45, the remainder [#permalink]
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Question Stats: Hide Show timer StatisticsWhen the positive integer n is divided by 45, the remainder is 18. Which of the following must be a divisor of n ?A. 11B. 9C. 7D. 6E. 4 Practice QuestionsQuestion: 10Page: 62 _________________
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Re: When the positive integer n is divided by 45, the remainder [#permalink]
sandy wrote: When the positive integer n is divided by 45, the remainder is 18. Which of the following must be a divisor of n ?A. 11B. 9C. 7D. 6 E. 4 We are given that when the positive integer n is divided by 45, the remainder is 18. We can express this in the remainder formula:n/45 = Q + 18/45 (where Q = Quotient)n = 45Q + 18Any divisor of n must evenly divide into 45Q and 18. Thus, 9 is the correct answer.Answer: B _________________
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Re: When the positive integer n is divided by 45, the remainder [#permalink] Divisibility Rule: DIVIDEND=(DIVISOR*QUOTIENT)+REMAINDER --(1) Given: 1) Dividend = 'n'.2)Divisor=45.3)Remainder=18.4)Quotient=?Let's assume the Quotient to be 'a',Putting the above values in the formula (1), we getn=(45*a)+18.n=45a+18.n=9(5a+2).Thus, n must be a multiple of 9.(or)When it comes to remainders, we have a nice rule that says: If N divided by D, leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc. For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc. Here, we are told that n divided by 45 leaves a remainder of 18, so the possible values of n are: 18, 63, 108,... etc. IMPORTANT: the question asks, "Which of the following must be a divisor of n? So, let's test the smallest possible value of n, which is 18, and check the answer choices. 18 is NOT divisible by 11, 7 or 4, so we can ELIMINATE A, C and E. So, the correct answer is either B or D Now test the next possible value of n, which is 63. 63 is NOT divisible by 6, so we can ELIMINATE DSo, by the process of elimination, the correct answer is B.
Retired Moderator Joined: 07 Jun 2014 Posts: 4806 GRE 1: Q167 V156
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Re: When the positive integer n is divided by 45, the remainder [#permalink] ExplanationThe given information tells you that n can be expressed in the form n = 45k + 18, where k can be any non-negative integer. Consider how the divisors of 45 and 18 may be related to the divisors of n. Every common divisor of 45 and 18 is also a divisor of any sum of multiples of 45 and 18, like 45k + 18. So any common divisor of 45 and 18 is also a divisor of n. Of the answer choices given, only 9 is a common divisor of 45 and 18. Thus the correct answer is Choice B. _________________
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Re: When the positive integer n is divided by 45, the remainder [#permalink] the 2nd explaination was more helpful then the first..
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Re: When the positive integer n is divided by 45, the remainder [#permalink]
sandy wrote: When the positive integer n is divided by 45, the remainder is 18. Which of the following must be a divisor of n ?A. 11B. 9C. 7D. 6 E. 4
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Re: When the positive integer n is divided by 45, the remainder [#permalink] Good approach
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Similar problem [#permalink]
Sid23 wrote: Good approach If k is the greatest positive integer such that 3^k is a divisor of 15! then k =A. 3B. 4C. 5D. 6E. 7
Retired Moderator Joined: 07 Jun 2014 Posts: 4806 GRE 1: Q167 V156
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Re: Similar problem [#permalink]
ruposh6240 wrote: Sid23 wrote: Good approach If k is the greatest positive integer such that 3^k is a divisor of 15! then k =A. 3B. 4C. 5D. 6E. 7 Post this as a new question on the forum, please. _________________
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Re: When the positive integer n is divided by 45, the remainder [#permalink] I think I'm being exceptionally stupid but I can't get my head around this as 63/ 9 = 7, there is no remainder? Surely a divisor of n shouldn't also be a divisor of 18 otherwise there would be no remainder?
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Re: When the positive integer n is divided by 45, the remainder [#permalink]
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Re: When the positive integer n is divided by 45, the remainder [#permalink] This one didn't really click with me until I tried one of the wrong answers. Let's try 11.N = 45Q + 18N = \(\frac{45Q + 18}{11}\) N = \(\frac{45Q}{11} + \frac{18}{11}\)You can clearly see now that N can not be an integer after division, which is what we want when looking for a divisor of N, unless BOTH 45Q and 18 are divisible by the answer choice.So, 11 wouldn't work.We can then try 9. 9 works. N will still be an integer if it is divided by 9. So, 9 is our answer.To illustrate:N = \(\frac{45Q}{9} + \frac{18}{9}\)N = 5Q + 2 . we can see N will be an integer if we divide by 9. So, 9 works.
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Re: When the positive integer n is divided by 45, the remainder [#permalink]
sandy wrote: ExplanationThe given information tells you that n can be expressed in the form n = 45k + 18, where k can be any non-negative integer. Consider how the divisors of 45 and 18 may be related to the divisors of n. Every common divisor of 45 and 18 is also a divisor of any sum of multiples of 45 and 18, like 45k + 18. So any common divisor of 45 and 18 is also a divisor of n. Of the answer choices given, only 9 is a common divisor of 45 and 18. Thus the correct answer is Choice B.
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Re: When the positive integer n is divided by 45, the remainder [#permalink]
sandy wrote: When the positive integer n is divided by 45, the remainder is 18. Which of the following must be a divisor of n ?A. 11B. 9C. 7D. 6E. 4 Practice QuestionsQuestion: 10Page: 62 When it comes to remainders, we have a nice rule that says:If N divided by D, leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc. For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.Here, we are told that n divided by 45 leaves a remainder of 18, so the possible values of n are: 18, 63, 108,... etc. IMPORTANT: the question asks, "Which of the following must be a divisor of n? So, let's test the smallest possible value of n, which is 18, and check the answer choices. 18 is NOT divisible by 11, 7 or 4, so we can ELIMINATE A, C and E. So, the correct answer is either B or DNow test the next possible value of n, which is 63. 63 is NOT divisible by 6, so we can ELIMINATE DSo, by the process of elimination, the correct answer is B _________________
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Re: When the positive integer n is divided by 45, the remainder [#permalink] When the positive integer n is divided by 45, the remainder is 18 Theory: Dividend = Divisor*Quotient + Remainder n -> Dividend45 -> Divisora -> Quotient (Assume)18 -> Remainders=> n = 45*a + 18 = 45a+18Which of the following must be a divisor of n n = 45a + 18=> n = 9*5a + 9*2=> n = 9*(5a + 2)=> n will be divisible by 9 and all factors of 9So, Answer will be B Hope it helps!Watch the following video to learn the Basics of Remainders _________________
Manager Joined: 27 Oct 2021 Posts: 59
When the positive integer n is divided by 45, the remainder [#permalink] As we clearly see 45 is not a divisor of n. Cuz to be a divisor it must completely divide the dividend which should lead us to remainder 0. But that's not the case here. We have a remainder of 18. It's a typical divisibility problem. I tackle divisibility problems in two ways. If there is a remainder use the remainder formula. That is Dividend = Divisor*Quotient + Remainder. If there is no remainder mentioned in the question use Dividend = Divisor*Quotient.Here Quotient is also a divisor of the dividend. Suppose we have a dividend of 14. So its divisors are 7*2. 14=7*2. Both are divisors of 14.
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When N is divided by 8 it leaves a remainder 3 and quotient R when N is divided by 5 leaves remainder 2 the value of N?
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