In ancient China, a general had 500 soldiers. After a war, he lost some of his s
ID: 2969569 • Letter: I
Question
In ancient China, a general had 500 soldiers. After a war, he lost some of his soldiers.
He estimated that he had at least 250 soldiers. He wanted to count the number of his remaining
soldiers. He asked his soldier to form groups of 17, i.e., each group consisted of 17 soldiers. There
were 12 soldiers who did not belong to any groups. He asked his soldier to form groups of 19, i.e.,
each group consisted of 19 soldiers. There were 2 soldiers who did not belong to any groups. How
many soldiers did the general have exactly now?
Explanation / Answer
Let the number of soldiers left after battle is N.
Take the largest of the two groups which is 19. When the total number of soldiers is divided by 19 gives a remainder of 2. Thus the number of soldiers is of the form: N=19k+2
SInce the same group of soldiers when divided into 17 each, we have a 12 soldiers who did not belong to any group , which is the remainder when N is divided 17.
We know that if there is a remainder in a division, by subtracting the remainder from the given number, the resulting number is divisible by the divisor. This means when 12 is subtracted from (19k+2), the resulting number i.e. (19k-10) is exactly divisible by 17.
Now give values of 0, 1, 2, 3 .... to k and find out for what value of k, (19k-10) is divisible by 17.
And the smallest value satisfying the above condition is k=5 which gives the number that we are looking for. Since the number, we said is (17k+12), the number of soldiers is (17*5)+12=97. So 97 is the smallest number which satisfes the two conditions.
The next higher number which satisifes this condition is obtained by adding LCM of (17,19) to the smallest number 97 found above.
LCM of 17 and 19 is 323.
Next number which satisfies the conditions is 323+97=420.
As he was counting the remaining soldiers, the count is 420.
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