There is an infinite number of positive integers answering the following definition:
each when divided by 3 leaves a reminder of 1,
when divided by 5 leaves a reminder of 3,
and when divided by 6 leaves a reminder of 4.
a. Find the smallest number with those features.
b. How many similar numbers below 1000 are there?
c. What is the largest integer below 10^6 with the above mentioned features?
Any number that leaves a remainder of 4 when divided by 6 also leaves a remainder of 1 when divided by 3, so we need worry only about division by 5 and by 6.
The first number leaving a remainder of 4 when divided by 6 is 10; then come 16, 22, 28, ....
The first of this arithmetic series to be 3 more than a multiple of 5 is 28, and so that's the answer to part a.
The LCM of 5 and 6 is 30, so every group of 30 consecutive numbers has one member. 1000 divided by 30 is 33 leaving a remainder of 10. The number 28 is not within the first 10 of the repeating cycle, so there are only 33 similar numbers below 1000, or rather there is 28 itself and 32 other similar numbers.
10^6 also leaves a remainder of 10 when divided by 30 so 10^6  10 is the last number of a full cycle of 30 and 2 before that is congruent to 28. Part c answer: 10^6  10  2 = 999988.

Posted by Charlie
on 20180825 15:45:55 