How many positive integers $n$ from 1 to 5000 satisfy the congruence $n \equiv 5 \pmod{12}$?
1 answer:
The equivalence 
means that n-5 is a multiple of 12.
that is
n-5=12k, for some integer k
and so
n=12k+5
for k=-1, n=-12+5=-7
for k= 0, n=0+5=5 (the first positive integer n, is for k=0)
we solve 5000=12k+5 to find the last k
12k=5000-5=4995
k=4995/12=416.25
so check k = 415, 416, 417 to be sure we have the right k:
n=12k+5=12*415+5=4985
n=12k+5=12*416+5=4997
n=12k+5=12*417+5=5009
The last k which produces n<5000 is 416
For all k∈{0, 1, 2, 3, ....416}, n is a positive integer from 1 to 5000,
thus there are 417 integers n satisfying the congruence.
Answer: 417
means that n-5 is a multiple of 12.
that is
n-5=12k, for some integer k
and so
n=12k+5
for k=-1, n=-12+5=-7
for k= 0, n=0+5=5 (the first positive integer n, is for k=0)
we solve 5000=12k+5 to find the last k
12k=5000-5=4995
k=4995/12=416.25
so check k = 415, 416, 417 to be sure we have the right k:
n=12k+5=12*415+5=4985
n=12k+5=12*416+5=4997
n=12k+5=12*417+5=5009
The last k which produces n<5000 is 416
For all k∈{0, 1, 2, 3, ....416}, n is a positive integer from 1 to 5000,
thus there are 417 integers n satisfying the congruence.
Answer: 417
You might be interested in
In 90 minutes the population to grow to 700 organisms.
Given that,
A strand of bacteria has a doubling time of 15 minutes.
If the population starts with 10 organisms.
We have to determine,
How long would it take for the population to grow to 700 organisms?
According to the question,
A strand of bacteria has a doubling time of 15 minutes.
If the population starts with 10 organisms.
In 15 minutes strand of bacteria has a doubling time the population starts with 10 organisms.
In 15 minutes 20 bacteria for the population to grow.
In 30 minutes strand of bacteria has a doubling time the population starts with 20 organisms.
In 30 minutes 40 bacteria for the population to grow.
In 45 minutes strand of bacteria has a doubling time the population starts with 40 organisms.
In 45 minutes 80 bacteria for the population to grow.
In 60 minutes strand of bacteria has a doubling time the population starts with 80 organisms.
In 60 minutes 160 bacteria for the population to grow.
In 75 minutes strand of bacteria has a doubling time the population starts with 160 organisms.
In 75 minutes 320 bacteria for the population to grow.
In 90 minutes strand of bacteria has a doubling time the population starts with 320 organisms.
In 90 minutes 640 bacteria for the population to grow.
In 120 minutes strand of bacteria has a doubling time the population starts with 640 organisms.
In 120 minutes 1280 bacteria for the population to grow.
Hence, In 90 minutes the population to grow to 700 organisms.
For more details refer to the link given below.