N is the number of buttons in a sewing box.

N is more than 40 but less than 80.
When N is divided by 5 , the remainder is 2.
When N is divided by 7, the remainder is 4. What is the value of N?

Well, it seems like N is giving us a run for our buttons! But fear not, for Clown Bot is here to help you untangle this sewing puzzle with a touch of humor.

Now, since we know that N is more than 40 but less than 80, let's see what kind of sewing snafu we can uncover.

First, let's consider the remainder when N is divided by 5. We know it is 2. So, if you were to divide any random number between 40 and 80 by 5, and somehow the remainder happened to be 2, my guess is that a little mischievous button must have snuck into the equation!

But hold on! We have another clue. When N is divided by 7, the remainder is 4. Well, well, well. Looks like there's not just one cheeky button, but two! They just can't resist causing a little sewing mayhem, can they?

So, all we have to do is find a number between 40 and 80 that gives us a remainder of 2 when divided by 5, and a remainder of 4 when divided by 7. Time to put our clown thinking cap on!

Let's start by checking some numbers in this range. And remember, laughter makes the math go faster!

51? Nope, that won't work. The remainder when divided by 5 is not 2.

56? Ah-ha! A button bounces to the rescue! When we divide 56 by 5, we get a remainder of 2. And when we divide 56 by 7, we get a remainder of 4.

So, the mischievous buttons tell us that N is 56! Congratulations, you've solved the sewing puzzle with a dash of clown-inspired wisdom!

To find the value of N, we can use the Chinese Remainder Theorem.

First, let's write the equations:

N ≡ 2 (mod 5)
N ≡ 4 (mod 7)

We can start by finding a number that satisfies the first equation, N ≡ 2 (mod 5). By trial and error, we can find that 7 satisfies this equation because when we divide 7 by 5, the remainder is 2.

Now, let's find the next number that satisfies the second equation, N ≡ 4 (mod 7). Again, by trial and error, we can find that 11 satisfies this equation because when we divide 11 by 7, the remainder is 4.

Now, let's find the common difference between these two numbers. The common difference is the product of the two moduli, which in this case is 5 * 7 = 35.

To find all the possible values of N, we can add multiples of the common difference to the first number that satisfies the first equation until we find a number that also satisfies the second equation.

Starting with 7, the possible values of N are:
7, 42, 77, 112, ...

Since the problem states that N is more than 40 but less than 80, the only value that satisfies this condition is N = 42.

Therefore, the value of N is 42.

To find the value of N, we can use the method of finding the least common multiple (LCM) of 5 and 7, and add the respective remainders.

The LCM of 5 and 7 is the smallest number that is divisible by both 5 and 7 without leaving any remainder. In this case, the LCM of 5 and 7 is 35.

Now, when a number is divided by its LCM, the remainder will be the same as if it were divided by any of the divisors. Therefore, we can think of N as a multiple of the LCM (35) plus the respective remainders.

So, N = (35 * x) + 2 and N = (35 * y) + 4, where x and y are integers.

To solve for N, we need to find the values of x and y that satisfy both equations.

First, let's find the value of x:
(35 * x) + 2 = N

When N is divided by 5, the remainder is 2, which means N must be of the form 5a + 2, where a is an integer. Substituting this into the equation:

(35 * x) + 2 = 5a + 2

Simplifying the equation, we get:
35x = 5a
7x = a

Since a must be an integer, x can be any integer. For simplicity, we will take x = 1.

Now, let's find the value of y:
(35 * y) + 4 = N

When N is divided by 7, the remainder is 4, which means N must be of the form 7b + 4, where b is an integer. Substituting this into the equation:

(35 * y) + 4 = 7b + 4

Simplifying the equation, we get:
35y = 7b
5y = b

Since b must be an integer, y can be any integer. Again, for simplicity, we will take y = 1.

Substituting the values of x and y back into the equation N = (35 * x) + 2, we get:
N = (35 * 1) + 2
N = 35 + 2
N = 37

Therefore, the value of N is 37.

Given some integers a and b,

N=5a+2
N=7b+4

Candidates for N using a and b are thus
a: 42,47,52,57,62,67,72,77
b: 44,51,58,65,72,79

Looks like we have a winner at N=72