When the positive integer n is divided by 45, the remainder is 18. Whi

\(n=45q+18=9(5q+2)\)

Thus n must be div by 9.

Answer: B

Solution

Given:
• When the positive integer n is divided by 45, the remainder is 18

To find:
• Out of the given options, which one must be a divisor of n

Approach and Working:
• When n is divided by 45, let’s assume that the quotient is k, with the remainder 18

o Therefore, we can write n = 45k + 18

• Now, irrespective of the value of k, n will be divisible by a number if 45 and 18 both are divisible by that number
• From the given options, only 9 can divide both 45 and 18

Hence, the correct answer is option B.

Answer: B
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least value of n=18
divisors include options 6,9; not sufficient
next least value of n=63
divisors include options 7,9; sufficient
9 must be divisor of n
B

We can find all possible values of n by adding the remainder 18 to integer multiples of 45. Thus, we see that n can be values such as 18 or 63 or 108.

The only common factors of these three numbers are 1 and 9. Since 1 is not in the answer choices, the correct answer must be 9.

Answer: B
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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 D

So, by the process of elimination, the correct answer is B

RELATED VIDEO FROM OUR COURSE

When the positive integer n is divided by 45, the remainder is 18

Theory: Dividend = Divisor*Quotient + Remainder

n -> Dividend
45 -> Divisor
a -> Quotient (Assume)
18 -> Remainders
=> n = 45*a + 18 = 45a+18

Which 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 9

So, Answer will be B
Hope it helps!

Watch the following video to learn the Basics of Remainders


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