M01-18

Question

How many positive three-digit integers are multiples of 7?

A. 105
B. 106
C. 127
D. 128
E. 142

Bunuel · Accepted Answer

Official Solution:

How many positive three-digit integers are multiples of 7?

A. 105
B. 106
C. 127
D. 128
E. 142

Generally, the number of multiples of x within a given range can be found using the formula:
 
 \frac{Last \ multiple \ of \ x \ in \ the \ range - First \ multiple \ of \ x \ in \ the \ range}{x} + 1 
 
For example, how many multiples of 5 are there between -7 and 35, not inclusive?
 
Last multiple of 5  IN  the range is 30;
 
First multiple of 5  IN  the range is -5;
 
 \frac{30 - (-5)}{5} + 1 = 8 . To demonstrate, the integers between -7 and 35 that are multiples of 5 are -5, 0, 5, 10, 15, 20, 25, and 30.
 
Or, How many multiples of 7 are there between -28 and -1, not inclusive?
 
Last multiple of 7  IN  the range is -7;
 
First multiple of 7  IN  the range is -21;
 
 \frac{-7 - (-21)}{7} + 1 = 3 . To demonstrate, the integers between -28 and -1 that are multiples of 7 are -21, -14, and -7.
 
 Back to the original question: 
 
Last 3-digit multiple of 7 is 994;
 
First 3-digit multiple of 7 is 105;
 
So, the number of 3-digit multiples of 7 is  \frac{994 - 105}{7} + 1 = 128 .

Answer: D

Mo2men · Answer

We have alternative approach with same line of logic presented by "Bunuel":

divide 1000 by 7 = 142.xxx It means we have 142 (No rounding up) numbers are multiple of 7 till 1000. For clarification if you round up, it means 143 * 7= 1001 and hence wrong

However, we asked about THREE Digit numbers ONLY. It means we need to find multiple of 7 till 100'

divide 100 by 7 = 14.xxxx It means we have 14 (No rounding up) numbers are multiple of 7

Total 3-digit numbers = 142-14=128

Answer: D

mvrravikanth · Answer

Hello,

I solved the question via A.P(Arithmetic progression)

The minimum three digit number divisible by 7 is 105 and maximum number is 994.
Hence,a=105,d=7,l=994
l=a+(n-1)d;
994=105+(n-1)7
by solving,we get
n=128.

Hope it helps!

Thanks.

DudeYou · Answer

I see most of the folks have quickly calculated 105 and 994 to be divisible by 7. I understand that its doable by simply checking numbers one by one, but is there an alternate way to do this ?
For example -- Find the first number divisible by 7 between 211 and 300 ?

Bunuel · Answer

210 = 7*30, so the next number divisible by 7 is 210 + 7 = 217.

itisSheldon · Answer

I have the same doubt as rohit. Approach 2 works all the time, there is no doubt on its credibility. But approach 1 is much more quicker than approach 2(since we dont need to find 1st and last terms). If the answer choices are not close we can safely use Approach 1(if 127 is not given as an option). Due to the glitches in approach 1, as explained by  kyles  in this thread, we cannot always use Approach 1.

chetan2u  /  Bunuel 
Please corroborate my opinion regarding approach 1

chetan2u · Answer

Hi...
ADDING 1 is an important aspect and can give you a wrong answer at times..
Rounding off is to lower INTEGER..
What are these DECIMALS- they give you the remainder..

Why adding 1 is important?
Say we have to find multiples of 10 in 3-digit numbers.
If you write 999-100=899.. divide by 10 --- 89.9, so 89..
But that's not correct.. Ans is 90.... 999-100+1=900

Now where can you go WRONG by using approacch 1..
When the numbers on either end are div by that number..
Say number of multiple of 10 between 100 and 200..
Approach 1 ..
200-100+1=101....
. 101/10=10.1~10
But multiples are 100,110,120,130,140,150,160,170,180,190,200=11
Here actually approach 1 becomes 2 and answer should be (200-100)/10 +1=10+1=11

AndrewN · Answer

Another approach : This one starts with the realization that there are 900 three-digit numbers. After that, mental math can get us the answer quickly and cleanly. It may not be evident how many times 7 goes into 900 evenly, but we should  know  that 7 goes into 700 exactly 100 times. With the bulk of the 900  numbers  out of the way, we just have to figure out how many times 7 goes into the remaining 200 numbers evenly. Rather than labor over the exact answer, we can see that 7 would fit into 210  exactly  30 times. But 700 (100 sevens) + 210 (30 sevens) = 910 (130 sevens), and we have no more than 900 numbers to work with instead. From 910, just subtract one 7 at a time until the number falls at or below 900: 910 - 7 = 903; 903 - 7 = 896. Since we had 130 sevens that fit into 910 numbers and needed to take away two sevens to give us a valid number, we know that 130 - 2 sevens, or 128 sevens, will be our answer.  (D) it is, then.  To recap:

* There are 900 three-digit numbers.

700/7=100  (sevens)

210/7=30  (sevens)

700+210=910  (100 + 30 = 130 sevens)

910-2(7)=896  (130 - 2 = 128 sevens)

This problem (or one similar to it, asking about a different divisor) can be broken down easily in under a minute, and nothing beats  knowing  your answer is correct.

Good luck with your studies.

- Andrew

EyeSeaMe · Answer

I completely concur with your method  AndrewN , it is a foolproof method. Folks, if you understand this method, then this method will also help you to gain confidence in the Number Properties topic.

I followed the same reasoning, with a slightly different approach as follows -

Last 2 digit number which is divisible by 7 = 98 (14 x 7)

First 3 digit number which is divisible by 7 = 105 (15 x 7) OR (98 + 7)

To find the last 3 digit number without multiple trials and errors, we know that it will be some number between 990 - 999

So, by splitting a value greater than 900 and between 990 - 999 that is divisible by 7, we get -

700 + 98 + 98 + 98 = 994 (Last number)
 
We can then solve the remaining part using  Bunuel  method OR we can reason as follows-

The last 3 digit number which is divisible by 7 = 994,

So 994/7 = 142. This means that the number 7 is added 142 times.

This also means that the first number in this series will be the number 7, which is a one-digit number as well as, other 2-digit numbers which are multiples of 7. But the question stem specifically asks for 3-digit numbers.

So, excluding all 1-digit and 2-digit numbers i.e numbers less than 100. There are 14 such numbers (14 x7 = 98).

Hence, 142 (Multiples of 7 less than 1000) - 14 (Multiples of 7 less than 100) = 128.

This reasoning looks difficult to comprehend at first, but it will definitely help us in getting better with our intuition while solving number property questions.

I hope it helps!

BottomJee · Answer

I think this is a  high-quality  question and I agree with explanation.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

MT1302 · Answer

I think this is a  high-quality  question and I agree with explanation.

ShenotShih1017 · Answer

There’s a formula for questions like this: #of multiples of X in a given range =   + 1

In this case  +1 = 128

If you didn’t know the formula this becomes much more difficult, but you could still divide 1000 by 7, and ignore the remainder to get the # of multiples of 7 <1000 = 142, then subtract the number of multiples of 7 <100 (14) to get 128.

Posted from my mobile device

vn049636 · Answer

I think this is a  high-quality  question and I agree with explanation.

GMAToverSanity · Answer

1. 7 ⋅ 15 = 105 (1st term) (just by knowing multiples of 15)
2. 999/7 = 142 w/ some irrelevant remainder ⇾ 7 ⋅ 142 = last term
3. 7 ⋅ 15, 7 ⋅ 16, 7 ⋅ 17, ... 7 ⋅ 142 are all of the 3 digit multiples of 7   inclusive  
4. 142-15   +1(bc/ inclusive)   =  128

satish_sahoo · Answer

Arithmetic Progression will also be an elegant solution here, we just need to realize that this will be a AP sequence

The set:  105,...,994

a_n=a+(n-1)d
994=105+(n-1)7
n=128

Borrat · Answer

Only on divisible by 7 is D : 12+ 2*8 = 28.

Zosima08 · Answer

I like the solution - it’s helpful.

akshayjain0927 · Answer

I like the solution - it’s helpful.

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