I am thinking of a number less than 60. When my number is divided by 4 it gives remainder 3 . When my number is divided by 5 it gives remainder 4 . When my number is divided by 6 it gives remainder 5
What number am I thinking of?
You are correct! Well done!
The number you are thinking of is 59.
To see why, we can work backwards from the given remainders:
- If the number gives a remainder of 5 when divided by 6, it must be one less than a multiple of 6. Possible options are: 5, 11, 17, 23, 29, 35, 41, 47, 53, and 59.
- Looking at the remainders when divided by 5, we see that the number must also be one more than a multiple of 5 (since the remainder is 4). The only option from the list above that satisfies this is 59.
- Finally, we can check that 59 gives a remainder of 3 when divided by 4, which confirms that it is the solution.
59
To solve this problem, we need to find a number that satisfies three conditions:
1. When divided by 4, it gives a remainder of 3.
2. When divided by 5, it gives a remainder of 4.
3. When divided by 6, it gives a remainder of 5.
We can solve this by finding the least common multiple (LCM) of the divisors in the conditions, which are 4, 5, and 6. The LCM of these numbers is 60.
So, the number we are looking for must be of the form x = 60n + k, where x is the number, n is an integer, and k is the remainder.
Now, we can substitute the remainders into the equation:
1. (60n + k) % 4 = 3
2. (60n + k) % 5 = 4
3. (60n + k) % 6 = 5
To find the number, we need to solve these equations simultaneously.
Let's start with the first equation:
(60n + k) % 4 = 3
To solve this equation, we can start by substituting some values for n and checking if the equation holds true. Let's start with n = 0:
(60 * 0 + k) % 4 = k % 4 = 3
Now, we need to find a value of k that satisfies this equation. We can try different values of k until we find one that works. Let's try k = 3:
3 % 4 = 3
Since k = 3 satisfies the equation, we can move on to the second equation:
(60n + k) % 5 = 4
Again, let's substitute n = 0 and k = 3 into the equation:
(60 * 0 + 3) % 5 = 3 % 5 = 4
This equation is also satisfied. Now, let's move on to the third equation:
(60n + k) % 6 = 5
Substituting n = 0 and k = 3:
(60 * 0 + 3) % 6 = 3 % 6 = 3
This value does not satisfy the equation. Let's try again with n = 1:
(60 * 1 + 3) % 6 = 63 % 6 = 3
Again, it does not satisfy the equation. Let's try n = 2:
(60 * 2 + 3) % 6 = 123 % 6 = 3
It still does not work. We need to keep trying different values of n until we find one that satisfies the third equation.
If we try n = 3:
(60 * 3 + 3) % 6 = 183 % 6 = 3
It works! Now, we have found the value of k that satisfies all three equations: k = 3.
Therefore, the number you are thinking of is x = 60n + k = 60 * 3 + 3 = 183.