find the least number which on being divided by5,6,8,9,12 leaves in each case a remainder 1?

Question

find the least number which on being divided by5,6,8,9,12 leaves in each case a remainder 1 but when divided by 13 leaves no remainder

Answer 1

if n is the number then n - 1 can be divided by 5 , 6, 8, 9 and 12.

5
6 = 2 * 3
8 = 2 * 2 * 2
9 = 3 * 3
12 = 2 * 2 * 3

the least common multiple therefore is:
5 * 2 * 2 * 2 * 3 * 3 = 360

So numbers which on being divided by 5, 6, 8, 9 and 12 leave a remainder 1 are
360 + 1 = 361 (leaves a remainder if divided by 13)
2*360 + 1 = 721 (leaves a remainder if divided by 13)
...
10*360 + 1 = 3601
This number leaves no remainder if divided by 13.

So the answer is 3601.

Answer 2

Let' s solve the first part first.
We will find the LCM of 5,6,8,9,12 and add 1
to the result to solve part 1.
Since 6 divides 12, it is enough to find the LCM
of 5 8 9 and 12.
Since 8 and 9 are relatively prime(i.e., have
no prime factor in common) their LCM
is their product or 72.
So we need the LCM of 5,72 and 12.
Since 12 divides 72 this is the LCM of
5 and 72. Finally, since 5 and 72 are
relatively prime, their LCM is 360.
So the answer to part 1 is 361.
Now we must find the smallest k such that
360k + 1 is divisible by 13.
By trial, we find that k = 10 and the answer is 3601.
Here's another way to do the last part:
We want
360k +1 = 0(mod 13)
360 k = 12(mod 13)
30k = 1(mod 13)
4k = 1(mod 13)
so
k = 10(mod 13)
and the answer is 3601.
Sorry for my earlier mistake!

Answer 3

Answer 3601 is correct which is given above!

Therefore, only a logical manner of arriving at said answer is explained below

5,6,8,9,12 are factors of 360 which is well known. Angle 360 degree is chosen because 360 is a number having greater equal-parts- divisibility (2 ,3 ,4 ,5 ,6 ,8, 9, 10, 12, 15, 18 ,20 , 24, 30, 36, 40, 45, 60, 72, 90, 120,180 etc...)

LCM method is a logical solution which has been done!

So 361 fulfils part 1 of question.

Part-2 is divisibility by 13, which you may continue as

361/13 = 27.769230769230769230769230769231

Remove whole number part of it and we get...

0.76923076923076923076923076923077

Reciprocal of its recurring digits is 1 / 0.769230
= 1.3000013000013000013000013000013

Find ratio 13/1.3 alone (by disregarding further recurring digits).

Answer is 10.

Multiply 360*10 which is 3600

To compensate effect of omitting recurring digits of number 1.3000013000013000013000013000013 in division 13/1.3 you can simply add '1' to 3600

So final answer is 3601.

It is always preferable to solve questions like these in a known manner and so I have explained it!


Regards!

Answer 4

With all of the injuries to the Cardinals starting pitching (Garcia, Westbrook) I would say there is a good chance it may be sooner than you think. How long can guys like Gast and Lyons fill the voids? Will Shelby Miller continue to be lights out? Do they fast track Wacha and call him up? It should be interesting. The Cards got off to a good start on their west coast trip but they will be facing a Dodger team fighting to save their manager and could be returning home in second place. The pitching has been phenomenal but is due to come back down to earth. I am objective what can I say.

Answer 5

Lcm of 5,6,8,9,12 is 360
so i have a no. A which is divisible by 13 but can also be expressed as
A=360k+1
k=natural no.
so i put k=10
and the no. is 360*7+1=3601

Hope its Right

Answer 6

361

Answer 7

361

Answer 8

181

Answer 9

Let n be the number
for some a,b,c,d,e
n = 5a+1
n = 6b+1
n = 8c+1
n = 9d+1
n=12e+1
let lcm = lowest common multiple
n= lcm(5.6.8.9.12)+1
= 180 + 1 = 181

Answer 10

Hey the answer is 3601