find the least number that when divided by 16 18 and 20 leaves a remainder 4 in case but is completely divisible by 7?
please give step by step explanation
Answers
Answer:
The Least number that when divided by 16, 18 and 20 leaves a remainder 4 in all case, but is completely divisible by 7 is 2884.
Step-by-step explanation:
Let N be the required number.
We should find a number that when divided by 16, 18 and 20 leaves an equal remainder of 4 in each case.
So, the number must be a common multiple of 16, 18 and 20.
Thus, we find its LCM
LCM (16, 18, 20) = 720
Now,
720 is divisible by 16, 18 and 20.
But it is only the lowest multiple of 16, 18 and 20, it might not necessarily be our number.
So,
We write the number as 720k, that will satisfy our needs.
Now,
720k is perfectly divisible by 16, 18 and 20, and gives remainder 0.
But we need a number that gives 4 as a remainder to all, 16, 18 and 20.
So, we simply add 4 to 720k = 720k + 4
Hence,
N = 720k + 4
Next the condition is that, when divided by 7, it must be perfectly divisible or in other words its remainder must be 0.
Thus,
N/7 = (720k + 4)/7
Now, if we try to divide 720 by 7 we get Quotient as 102 and remainder as 6
So,
We can write 720k + 4 as (714k + 6k + 4)
We can perfectly divide 714 as it is divisible by 7.
714 ÷ 7 = 102
So,
N/7 = (714k + 6k + 4)/7
= (714k/7) + (6k/7) + (4/7)
= 102k + (6k + 4)/7
Hence,
Now comes the final procedure, 714k has been divided successfully, now only 6k + 4 remains.
6k + 4 must be a multiple of 7, only then can we divide it by 7 and get remainder 0
So,
we need to adjust 'k' so that, it becomes a multiple of 7, or it must be divisible by 7.
Next is a trial and error method.
When k = 1
6(1) + 4 = 10
10 is not a multiple of 7.
k = 2,
6(2) + 4 = 16
16 is not a multiple of 7.
k = 3
6(3) + 4 = 22
22 is not a multiple of 7.
k = 4
6(4) + 4 = 28
28 is a multiple of 7.
Hence,
k = 4 gives the lowest number which is divisible by 7.
So,
N = 720(4) + 4
N = 2880 + 4
N = 2884
OR
N/7 = (720k + 4)/7
Now, we simply put in values for k till it becomes divisible by 7.
When k = 1
720(1) + 4 = 724
724 is not divisible by 7.
k = 2,
720(2) + 4 = 1444
1444 is not divisible by 7.
k = 3
720(3) + 4 = 2164
2164 is not divisible by 7.
k = 4
720(4) + 4 = 2884
2884 is divisible by 7.
Hence,
N = 2884
The first method is better as the multiplication and division procedures could be time consuming in the second method, whereas in the first method, it deals with smaller numbers.
Now, we can check if it is correct also,
2884 ÷ 16 = 180 and Remainder = 4
2884 ÷ 18 = 160 and Remainder = 4
2884 ÷ 20 = 144 and Remainder = 4
2884 ÷ 7 = 412 and Remainder = 0, which means 7 completely divides 2884.
Hence,
N = 2884
The Least number that when divided by 16, 18 and 20 leaves a remainder 4 in all case, but is completely divisible by 7 is 2884.
Hope it helped and believing you understood it........All the best