If R is the remainder when each of the numbers 1059, 1417, and 2312 is divided by D, where D is an integer greater than 1, compute the value of D-R
This is a very challenging question, and a very brilliant answer by Reiny.
I would attempt the problem the following way. Let me know if it is correct.
We are given three numbers with an unknown offset. The three (original) numbers have a common factor. We are to find the common factor and the unknown offset.
We can find the unknown common factor by Euclid's method, which is to take the difference of the three numbers in turn and find the HCF of the differences.
1417-1059=358
2312-1417=895
D=HCF(895,358)=179
Check:
1059/179=5 R 164
1417/179=7 R 164
2312/179=12 R 164
D-R=179-164=15 same answer as Reiny.
D = 179, R = 164
so d-r = 179-164 = 15
15
GDJF
=15
To find the remainder when a number is divided by D, we need to use the modulus operator (%). We can calculate the remainder for each number individually, using the formula "number % D". Let's calculate the remainders for 1059, 1417, and 2312 step by step:
Remainder of 1059 divided by D: 1059 % D = R1
Remainder of 1417 divided by D: 1417 % D = R2
Remainder of 2312 divided by D: 2312 % D = R3
Now, we need to compute the value of D-R, which is (D - R1) + (D - R2) + (D - R3). Let's simplify this expression:
D - R1 + D - R2 + D - R3
= 3D - (R1 + R2 + R3)
We know that the sum of the remainders should be equal to the sum of the original numbers:
R1 + R2 + R3 = 1059 + 1417 + 2312
= 4788
Therefore, the expression simplifies to:
3D - (R1 + R2 + R3)
= 3D - 4788
So, the value of D - R would be 3D - 4788.