M01-18

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.

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 ?

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

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

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

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

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!

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

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

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