How many numbers between 1 to 300 are divisible by 11 and 13 but not

hi
good question
i would start with listing all the multiples of 11 and 13
11
22
33
44
55
66
77
88
99
110
121
132
143 - multiple of 13 too
154
165
176
187
198
209
220
231
242
253
264
275
286 - another multiple of 13
297



multiples of 13
13
26
39
52
65
78
91
104
117
130
143 - multiple of 11
156
169
182
195
208
221
234
247
260
273
286 - multiple of 11
299


total 46
answer is B.

alternatively, we can divide 300 by 11 and 300 by 13 and see how many multiples are for each.
300/13 = 23 and smth, so 23 multiples
300/11 = 27 and smth, so 27 multiples

total - 50, but out of these, 11*13 should be removed (-2 numbers because it counts once in multiples of 11 and once in multiples of 13), and 22*13 should be removed (-2 numbers because it counts once in multiples of 11 and once in multiples of 13)
50-4 = 46
answer is B.

Dear annusngh,

I'm happy to respond. We can solve this without listing.

Start with multiples of 11:
there are ten up to and including 110
there are twenty up to and including 220
that leaves us a gap of 80
we could fit 7 more multiples of 11, which would take up 7*11 = 77 of that gap, ending at 220 + 77 = 297
that's 27 multiples of 11 less than 300

Now, multiples of 13:
there are ten up to and including 130
there are twenty up to and including 260
that leaves us a gap of 40
we could fit three more multiples of 13, which would take up 3*13 = 39 of that gap, ending at 260 + 39 = 299
that's 23 multiples of 13 less than 300

Now, the overlaps.
13*11 = 13*(10 + 1) = 130 + 13 = 143
Double that:
286
If we take any higher multiples of 143, clearly we'll be over 300. We have to subtract those two numbers from both lists.

number = (27 - 2) + (23 - 2) = 25 + 21 = 46

answer = (B)

Does all this make sense?
Mike
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mikemcgarry, Why we need to subtract the overlap instead of adding up the overlap since 143 and 286 are less than 300?
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Thank you! If anyone as a faster way to do it!

Hi

A faster method will be..
Number of factors of 11=300/11=27.xyz...
Neglect decimals. There are 27 multiples of 11..

Number of factors of 13=300/13=23.xyz..
Again 23 factors..

Common factors 11*13=143 and 143*2...
So 2 factors..

Total factors=27+23-2(2)=46

B

Important:-We subtract the 2 factors twice because they have been added twice , once as a multiple of 11 and then as a multiple of 13..
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Hi Chetan,

A. Yes, this has only one trap of double counting of common multiples.
B. Question is so simple that it allures us to use simple counting approach of uncommon factors which is lengthy one. And that lengthy approach leads us to the attention deficiency and some of us miss the double deduction.

I think this can be solved through Arithematic Progression too. using the nth term of an AP is a(n)= a(1)+(n-1)d

Number of multiples of 11

297 = 11+(n-1)11
Solving for n we get n = 27

Similarly,

299 = 13+(n-1)13
n = 23

Now,divisible both by 11*13

286 = 143+(n-1)11*13
Solving we get n = 2

Total number of multiples = (27-2)+(23-2)= 46

We can first determine the number of multiples of 11 between 1 and 300 inclusive using the following formula:

(largest multiple of 11 - smallest multiple of 11)/11 + 1

(297 - 11)/11 + 1 = 286/11 + 1 = 26 + 1 = 27

Next we can determine the number of multiples of 13 between 1 and 300 inclusive:

(largest multiple of 13 - smallest multiple of 13)/13 + 1

(299 - 13)/13 + 1 = 286/13 + 1 = 22 + 1 = 23

Finally, we need to subtract out the numbers that are divisible by both 11 and 13. Since the least common multiple of 11 and 13 is 143, we see that 143 and (2 x 143) = 286 are divisible by both 11 and 13.

From the 27 multiples of 11, we eliminate the values 143 and 286, so we have 25 numbers remaining. Similarly, from the 23 multiples of 13, we also eliminate the values 143 and 286, so we have 21 numbers remaining.

Thus, the number of multiples of either 11 or 13 between 1 and 300 but not divisible by both 11 and 13 is equal to 25 + 21 = 46.

Answer: B
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annusngh

Multiples of 11 = 11, 22, 33, ....., 297 (forms a progression with first term = 11 and common difference = 11)
total = (297-11) / 11 + 1 = 27

Multiples of 13 = 13, 26,......,299 (forms a progression with first term = 13 and common difference = 13)
total = (299-13) / 13 + 1 = 23

Multiples of 13,11 = multiples of LCM (11,13) = 11*13 = 143 ...... only 2 are present (143 and 286)

So ANS: 27 + 23 - 2*2 = 46 ... so, C.

Can someone explain why do we need to do 50-4 instead of 50-2? only 143 and 286 are divisible by 11 and 13 together right?

There are 27 multiples of 11.
There are 23 multiples of 13.
There are 2 multiples of 143 (of both 11 and 13).

Note that the 27 numbers and the 23 numbers from above, both have these 2 numbers included. But you want the numbers which are divisible either by 11 or by 13 BUT NOT BY BOTH.
So there are 27 - 2 = 25 numbers divisible by 11 only.
And there are 23 - 2 = 21 numbers divisible by 13 only.

In all, there are 25 + 21 = 46 numbers divisible by 11 or 13 but not both.

Answer (B)
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Numbers between 1 to 300 are divisible by 11 = 300/11 => 27
Numbers between 1 to 300 are divisible by 13 = 300/11 => 23
Numbers between 1 to 300 are divisible by 11& 13 = 300/143 => 2





As you can see we are subtracting 2 from both 27 & 23 , thus we are subtracting 2 from the result of 27 & 23..

Hope this helps now!!

Answer will hence be (B) 46
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divisible by 11, divide 300 by 11 we get quotient 27
Divisible by 13, divide 300 by 13 we get quotient 23
out of 27 multiples 143 and 286 are also multiple of 13 so no of multiples of 11 and not 13= 25
out of 23 multiples 143 and 286 are also multiple of 11 so no of multiples of 13 and not 11= 21
therefore 25+21 = 46 ans

Multiples of 13 in 300= 300/13 => 23
Multiples of 11 in 300= 300/11 => 27
Multiples of 143(11*13) in 300= 300/143 => 2 (143 and 286)
since common is added to individual results of both 11 and 13, we'll remove it from each of them:
23+27-2-2 = 46
Answer - Option B
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To organize the data, use the following matrix:


Multiples of 11 between 1 and 300 = 300/11 ≈ 27
Multiples of 13 between 1 and 300 = 300/13 ≈ 23
Since 11*13 = 143 and 2*143 = 286, multiples of both 11 and 13 between 1 and 300 = 2
Inserting these values into the matrix, we get:

In the resulting matrix, (only 11) + (only 13) = 25 + 21 = 46


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