How many multiples of 33 lie between 101 and 1,000, inclusive?

Question

How many multiples of  33  lie between  101  and  1,000,  inclusive?
 
 A. 24  
 B. 27  
 C. 33  
 D. 36  
 E. 48

KSBGC · Accepted Answer

First multiple of 33 in the range : 132 (33*4)

Last Multiple in the range : 990 (33*30)

No. of multiple in the range : (30-4) + 1 = 27.

The best answer is B.

BrentGMATPrepNow · Answer

Some positive multiples of 33 are: 33, 66, 99,  132 , 165, 198,. . . , 957,  990 , 1023
So, we want the number of multiples of 33 from  132  to  990  inclusive

Observe: 
  132  = (33)( 4 )
165 = (33)(5)
198 = (33)(6)
.
.
.
957 = (33)(29)
 990  = (33)( 30 )

We can see that the number of multiples of 33 from  132  to  990  inclusive is the SAME as the number of integers from  4  to  30  inclusive.

To determine the above, we can apply the following rule:   the number of integers from x to y inclusive equals y - x + 1  
So, the number of integers from  4  to  30  inclusive =  30  -  4  + 1 = 27

Answer: B

Cheers,
Brent

BrentGMATPrepNow · Answer

Another approach is to apply the following rule:

If x and y are multiples of k, then the number of multiples of k from x to y inclusive =   + 1  
So, for example, the multiples of 3 from 6 to 21 inclusive =   + 1 =   + 1 =  6

So, the number of multiples of 33 from  132  to  990  inclusive = ( 990  -  132 )/33 + 1
= 858/33 + 1
= 26 + 1
= 27

Answer: B

Cheers,
Brent

GMATGuruNY · Answer

Since 1000 - 101 ≈ 900 -- and the answer choices are a bit spread out -- we can count the multiples of 33 simply by dividing 33 into 900: 
900/33 = 300/11 = a bit more than 27. 
Thus, there are 27 multiples of 33 between 101 and 1000.

B

fskilnik · Answer

One of our students´ most-loved mottos is: let the "Queen of Sciences" (Mathematics) do the weight-lifting for you!

101 < 33M < 1000

? = M\,\,\,\left( {\operatorname{int} } \right)

\left( {99 + 33 = } \right)\,\,\,132 \leqslant 33M \leqslant 990\,\,\,\,\left( { = 30 \cdot 33} \right)

4 \leqslant M \leqslant 30\,\,\,\, \Rightarrow \,\,\,\,? = 30 - 4 + 1 = 27

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

Regards, 
fskilnik.

P.S.: that´s EXACTLY Selim´s solution, only a bit more "structured". Congrats, Selim!!

MathRevolution · Answer

=>
Consider the arithmetic sequence  132, 165, …, 990  of multiples of  33 .
The number of terms in this sequence is  \frac{(990-132)}{33} + 1 =\frac{858}{33} + 1 = 26 + 1 = 27.

Therefore, the answer is B.
Answer: B

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How many multiples of 33 lie between 101 and 1,000, inclusive?

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How many multiples of 33 lie between 101 and 1,000, inclusive?

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