find the largest number that will divide 220,313,and 716 leaving remainder 3 in each case?
217 = 7*31
310 = 10*31
713 = 23*31
It appears that dividing any of the original numbers by 31 will leave a remainder of 3.
To find the largest number that divides three given numbers (220, 313, and 716) and leaves a remainder of 3 in each case, we can use the concept of the greatest common divisor (GCD).
The GCD is the largest positive integer that divides two or more numbers without leaving a remainder. In this case, we are looking for the largest number that leaves a remainder of 3 when dividing these three numbers.
To find the GCD, we can use the Euclidean algorithm. Here's how to do it step by step:
1. Start by taking the first two numbers and finding their GCD. In this case, we need to find the GCD of 220 and 313.
2. Divide the larger number by the smaller number and find the remainder. In this case, 313 divided by 220 equals 1 with a remainder of 93.
3. Now, take the smaller number (which is 220) and the remainder from the previous step (which is 93) and find their GCD. In this case, we need to find the GCD of 220 and 93.
4. Repeat steps 2 and 3 until the remainder becomes 0. In this case, the next step would be to find the GCD of 93 and the remainder when dividing 220 by 93.
5. Finally, when the remainder becomes 0, the GCD will be the last non-zero remainder found. In this case, the GCD of 220 and 93 is 31.
Therefore, the largest number that will divide 220, 313, and 716, leaving a remainder of 3 in each case, is 31.
31
220 - 3 = 217
313 - 3 = 310
716 - 3 = 713
Find the prime factorization of each integer.
217 = 7 ∙ 31
310 = 2 ∙ 5 ∙ 31
713 = 23 ∙ 31
The "Greatest Common Factor " is the largest of the common factors (of two or more numbers)
In this case GCF = 31
Because 31 is the greatest number that divides evenly into all of them.
The largest number that will divide 220,313,and 716 leaving remainder 3 = 31
220 / 31 = ( 217 + 3 ) / 31 = 217 / 31 + 3 / 31 = 7 + 3 / 31
A remainder = 3
313 / 31 = ( 310 + 3 ) / 31 = 310 / 31 + 3 / 31 = 10 + 3 / 31
A remainder = 3
716 / 31 = ( 713 + 3 ) / 31 = 713 / 31 + 3 / 31 = 23 + 3 / 31
A remainder = 3