The number is successively divided by 5, 7, 11 give remainder as 3, 1, 10.
If last quotient is 114.
Find the numbers.
I will attempt the simplest approach:
numbers which when divided by 5 leave a remainder of 3 :
8, 13, 18, 23, 28, 33, 38, 43, 48, 53, 58, ...
numbers which when divided by 7 leave a remainder of 1
8, 15, 22, 29, 36, 43, 50, 57, 64, ....
numbers which when divided by 11 leave a remainder of 10
10, 21, 32, 43, 54 ...
ahhh, notice 43 satisfies all 3 conditions.
This was lucky, if the number had been larger, this method would become
impractical. In that case I had hinted in another reply that I would use
"The Chinese Remainder Theorem"
btw, if we multiply the divisors, and then add or subtract multiples of
that product we get another number that satisfies.
so 5*7*11 = 385
which means 43+385 or 428 would also work , you can check it !
so possible answers are 43 + k(385) , where k is a whole number.
Sorry dear
Didn't understand
Kindly explain in another way
Please
I don't understand that you "don't understand"
I gave you the simplest and most direct way of doing this.
I repeat my simple way:
I will attempt the simplest approach:
numbers which when divided by 5 leave a remainder of 3 :
8, 13, 18, 23, 28, 33, 38, 43, 48, 53, 58, ...
numbers which when divided by 7 leave a remainder of 1
8, 15, 22, 29, 36, 43, 50, 57, 64, ....
numbers which when divided by 11 leave a remainder of 10
10, 21, 32, 43, 54 ...
ahhh, notice 43 satisfies all 3 conditions.
It said find the numbers, I assumed you wanted more than just the 43.
That's where the last part comes in.
To find the numbers, we need to work backwards from the given information. Let's go step by step:
Step 1: Determine the remainder when the number is divided by 11.
We are given that the remainder when the number is divided by 11 is 10.
Step 2: Determine the remainder when the quotient from Step 1 is divided by 7.
We are given that the remainder when the number divided by 7 is 1.
Step 3: Determine the remainder when the quotient from Step 2 is divided by 5.
We are given that the remainder when the number divided by 5 is 3.
Step 4: Find the number using the given information.
Let's call the number we are trying to find "x."
Using the information from Step 4, we can set up the following equations:
x ≡ 3 (mod 5) (equation 1)
x ≡ 1 (mod 7) (equation 2)
x ≡ 10 (mod 11) (equation 3)
Now let's solve these equations step by step:
Step 5: Solve equation 1.
Since x ≡ 3 (mod 5), the possible values of x are 3, 8, 13, 18, ...
Step 6: Solve equation 2.
Since x ≡ 1 (mod 7), the possible values of x are 1, 8, 15, 22, ...
Step 7: Solve equation 3.
Since x ≡ 10 (mod 11), the possible values of x are 10, 21, 32, 43, ...
Step 8: Find the common value of x that satisfies all three equations.
From the possible values of x obtained from equations 1, 2, and 3, we can observe that x = 8 satisfies all three equations.
So, the number is 8.
Finally, since the last quotient is given as 114, we can say that the numbers are 8 and 114.