Question:
Find the greatest number that will divide 445, 572 and 699 leaving remainders 4, 5 and 6 respectively.
Solution:
Find the greatest number that divides 445, 572 and 699 and leaves remainders of 4, 5 and 6 respectively.
The required number when divides 445,572 and 699 leaves remainders 4,5 and 6 this means $445-4=441,572-5=567$ and $699-6=693$ are completely divisible by the number.
Therefore, the required number = H.C.F. of 441, 567 and 693.
First consider 441 and 567.
By applying Euclid’s division lemma
$567=441 \times 1+126$
$441=126 \times 3+63$
$126=63 \times 2+0$
Therefore, H.C.F. of 441 and 567 = 63
Now, consider 63 and 693
By applying Euclid’s division lemma
$693=63 \times 11+0$
Therefore, H.C.F. of 441, 567 and 693 = 63
Hence, the required number is 63