Find the greatest number which divides 2011 and 2623 , leaving remainders 9 and 5 respectively
Student
Life
To find the greatest number that divides two given numbers and leaves specific remainders, we can use the method of synthetic division.
Step 1: Start by finding the difference between the two remainders: 9 - 5 = 4
Step 2: Set up the equation using the difference as the divisor:
(x - 4)
Step 3: Now, use synthetic division to check if the given numbers leave the specified remainders when divided by (x - 4).
For 2011:
- Write 2011 as the dividend inside the synthetic division bracket, and write the coefficient of the divisor outside the bracket:
4 | 2011
- Start by bringing down the leading coefficient (in this case, the 2):
4 | 2011
|_____
|₂₀₁₁
- Multiply the divisor (4) by the result (2), and place the product below the next term of the dividend:
4 | 2011
| 8
|_____
|₂₀₁₁
|₁₆₀₈
- Subtract the product from the dividend:
4 | 2011
| 8
|_____
|₂₀₁₁
|₁₆₀₈
|４３
The remainder is 43.
For 2623:
- Write 2623 as the dividend inside the synthetic division bracket, and write the coefficient of the divisor outside the bracket:
4 | 2623
- Start by bringing down the leading coefficient (in this case, the 2):
4 | 2623
|_____
|₂₆₂₃
- Multiply the divisor (4) by the result (2), and place the product below the next term of the dividend:
4 | 2623
| 8
|_____
|₂₆₂₃
|₁₆₈
- Subtract the product from the dividend:
4 | 2623
| 8
|_____
|₂₆₂₃
|₁₆₈
|９５
The remainder is 95.
Step 4: If the remainders obtained during the synthetic division for both numbers are the same, it means (x - 4) is a common factor of 2011 and 2623.
Step 5: Further factorize the common factor, (x - 4), to obtain the greatest number that divides both numbers and leaves the specified remainders.
(x - 4) = 0
Solving for x, we get:
x = 4
Therefore, the greatest number that divides 2011 and 2623, leaving remainders 9 and 5 respectively, is 4.
Subtract 9 and 5 respectively.
2011 - 9 = 2002, try the smallest divisor to give you the greatest.
Do the same with 2623.