1,
:: LCM = 2 x 2 x 2 x 2 x 2 x 3 x 3 x 5 = 1440
Now, least number of 5 digits = 10000
We find that when 10000 is
1440)10000 6
divided by 1440, the
8640
remainder is 1360.
1360
Least number of 5 digits
which is exactly divisible by 1440 (ie. LCM of
32, 36, 40, 45 and 48) - 10000 + (1440 - 1360)
- 10000 + 80
10080
.
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Answers
Answer:
Since the desired number leaves remainder 27 when divided by 10, 12, 16, 18.
Therefore, we can say that if we subtract 27 from the required number, the resultant number will be a common multiple of 10, 12, 16, 18. Also this number will be a multiple of the LCM( Least Common Multiple ) of these numbers
Let the desired number be x
Firstly we find the LCM
10 = 2 x 5
12 = 2 x 2 x 3
16 = 2 x 2 x 2 x 2
18 = 2 x 3 x 3
LCM will be 2 x 2 x 2 x 2 x 3 x 3 x 5 = 720
Now we need to find multiples of 720 which are closest to 10000(the first 5-digit number)
The numbers closest to 10000 and are multiple of 720 are 9360 and 10080
Among these two numbers, when 27 is added to 9360, it does not give us a 5 digit number, whereas the second number does.
So we know x - 27 = 10080
x = 10107
Therefore 10107 is the smallest natural number which leaves remainder 27, when divided by 10, 12, 16, 18.
Explanation:
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