A number has 3 digits. When it is divided by 6 or 7, it leaves a remainder of 1. When it is divided by 8 or 11, it leaves 2 remainder of 7.
What is the largest such number?
The answer given is 799
I suggest you Google something like
"Chinese Remainder Theorem" and I assume you know about modular arithmetic since I will be using properties of modular arithmetic
so we have
z = 1 mod 6
z = 1 mod 7
z = 7 mod 8
z = 7 mod 11
But, in the Chinese Remainder Therorem we need all the modular arguments to be "relatively prime"
6 and 8 are not
BUT, 1 mod 6 = 7 mod 6
and if 7 mod 6 = 7 mod 8
then 7 mod 6 = 7 mod 24 , (LCM of 6 and 8 is 24)
so we can reduce our problem to 3 conditions:
z = 7 mod 24
z = 1 mod 7
z = 7 mod 11
BUT 7 mod 11 = 7 mod 24
so we now have
z = 7 mod 264
z = 1 mod 7
starting with the smallest 3 digit number, results of
7 mod 264 are
271 539 799 1063 ...
only 3 cases to check for mod 7
271 mod 7 = 5
539 mod 7 = 0
799 mod 7 = 1 ======> yeahhh, its 799
NOTE:
I did not actually have to use the "Chinese Remainder Theorem" since I was fortunate to have 7 mod 24 = 7 mod 11
If I had ended up with something like
5 mod 24 and 6 mod 11, it would be much more difficult and I would have to resort to the CRT
To find the largest number that satisfies the given conditions, we need to start by finding the least common multiple (LCM) of 6, 7, 8, and 11 because the number should leave a certain remainder when divided by each of these numbers.
1. Find the LCM of 6, 7, 8, and 11:
To find the LCM, list the multiples of each number until you find a common multiple.
Multiples of 6: 6, 12, 18, 24, 30,...
Multiples of 7: 7, 14, 21, 28, 35,...
Multiples of 8: 8, 16, 24, 32, 40,...
Multiples of 11: 11, 22, 33, 44, 55,...
From the lists above, we can see that 24 is the smallest number that all four numbers divide evenly into.
2. Find the remainder of dividing the number 24 by 6, 7, 8, and 11:
When 24 is divided by 6, the remainder is 0.
When 24 is divided by 7, the remainder is 3.
When 24 is divided by 8, the remainder is 0.
When 24 is divided by 11, the remainder is 2.
3. Find the largest number less than 24 that leaves a remainder of 1 when divided by 6 or 7:
From the given conditions, we need to find a number that leaves a remainder of 1 when divided by both 6 and 7. The largest number less than 24 that satisfies this condition is 6 * 7 + 1 = 43.
4. Find the largest number less than 24 that leaves a remainder of 7 when divided by 8 or 11:
From the given conditions, we need to find a number that leaves a remainder of 7 when divided by both 8 and 11. The largest number less than 24 that satisfies this condition is 8 * 11 + 7 = 95.
5. Find the largest number less than 24 that satisfies both conditions:
The largest number that satisfies both conditions is the smaller of the two numbers found in steps 3 and 4, which is 43.
Therefore, the largest number that satisfies the given conditions is 799.