Question
Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3 respectively.
✓
Answer
Since, 1,2 and 3 are the remainders of 1251, 9377 and 15628, respectively.
Thus, after subtracting these remainders from the numbers.
We have the numbers, 1251 - 1 = 1250, 9377 - 2 = 9375 and 15628 - 3 = 15625 which is divisible by the required number.
Now, required number = HCF of 1250, 9375 and 15625 [for the largest number]
By Euclid’s division algorithm,
a = bq + r .....(i)
[$\because$ dividend = divisor × quotient + remainder]
For largest number, put a = 15625 and b = 9375
15625 = 9375 × 1 + 6250
⇒ 9375 = 6250 × 1 + 3125
⇒ 6250 = 3125 × 2 + 0
$\therefore$ H.C.F (15625, 9375) = 3125
Now, we take c = 1250 and d = 3125, then again using Euclid's division algorithm,
d = cq + r [from Eq. (i)]
⇒ 3125 = 1250 × 2 + 625
⇒ 1250 = 625 × 2 + 0
$\therefore$ H.C.F (1250, 9375, 15625) = 625
Hence, 625 is the largest number which divides 1251, 9377 and 15628 leaving remainder 1, 2 and 3, respectively.
Thus, after subtracting these remainders from the numbers.
We have the numbers, 1251 - 1 = 1250, 9377 - 2 = 9375 and 15628 - 3 = 15625 which is divisible by the required number.
Now, required number = HCF of 1250, 9375 and 15625 [for the largest number]
By Euclid’s division algorithm,
a = bq + r .....(i)
[$\because$ dividend = divisor × quotient + remainder]
For largest number, put a = 15625 and b = 9375
15625 = 9375 × 1 + 6250
⇒ 9375 = 6250 × 1 + 3125
⇒ 6250 = 3125 × 2 + 0
$\therefore$ H.C.F (15625, 9375) = 3125
Now, we take c = 1250 and d = 3125, then again using Euclid's division algorithm,
d = cq + r [from Eq. (i)]
⇒ 3125 = 1250 × 2 + 625
⇒ 1250 = 625 × 2 + 0
$\therefore$ H.C.F (1250, 9375, 15625) = 625
Hence, 625 is the largest number which divides 1251, 9377 and 15628 leaving remainder 1, 2 and 3, respectively.
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