Question

Find the greatest number that will divide 138, 183 and 243 leaving the remainder 3 in each case.

Answer 1

Hint: Now we are given with three numbers 138, 183 and 243. We will first subtract the numbers with 3. Now we will find the factors of the numbers obtained and then find the highest common factor. The Highest common factor that we get is the required number.

Complete step by step answer:
Now the given three numbers are 138, 183 and 243.
Let us say x is the greatest number that will divide 138, 183 and 243 leaving the remainder 3.
Now we know that if x divides 138, 183 and 243 leaving the remainder 3 then, x will divide 138 – 3, 183 – 3 and 243 – 3.
Hence x divides 135, 180 and 240.
Hence we have x is the greatest number which divides 135, 180 and 240.
Now let us first find the factors of each of the following numbers.
135 = 3 * 3 * 3 * 5 * 5
180 = 2 * 2 * 3 * 3 * 5
240 = 2 * 2 * 2 * 2 * 3 * 5
Hence we can see that the maximum common factor that we get here is 5 * 3 = 15.
Hence the greatest number which divides 135, 180 and 240 is 15.

Hence we get the greatest number which divides 138, 183 and 243 and leaves remainder 3 is 15.

Note: Note that when we say x divides 138 and leaves remainder 3 this means we can write 138 as x.q + 3. Which is nothing but the form dividend = divisor * quotient + remainder. Now since we have 138 = x.q + 3 this means 138 – 3 = x.q. or 135 is divisible by x. This is the concept we have used to find the required number.