Question

Three numbers which are coprime to each other are such that the product of the first two numbers is 551 and the product of the last two numbers is 1073. The sum of the three numbers is
[a] 85
[b] 75
[c] 65
[d] 55

Answer 1

Hint: Use the property that if three numbers a,b,c are mutually coprime, then HCF(ab,bc) = b. Use Euclid's division lemma to find the HCF of 551 and 1073. Hence find the value of b. Now a can be obtained from ab by dividing it by b and c can be obtained by bc by dividing it by b. Hence the three numbers can be determined.

Complete step-by-step solution -

Let the three numbers be a,b and c.
Since a, b and b,c are coprime, we have HCF(a,b) = 1, HCF(a,c) = 1 and HCF(b,c) = 1.
Now we know that if HCF(a,b) = k, then HCF(ca,cb) =kc.
Since HCF(a,c) = 1, we have HCF(ba,bc) = b.
Given that product of the first two numbers is 551
Hence we have ab = 551.
Also, the product of the last two numbers is 1073.
Hence we have bc = 1073.
Euclid's Division Algorithm:
Finding HCF using Euclid's division lemma is an algorithmic process. We start by setting the two variables p and q equal to the given numbers, with q being equal to the smaller one. Then we repeat the following process:
Apply Euclid's division lemma to the numbers p and q
i.e. p = aq+r where $0 \le r < q$
if r = 0 then stop the process and we have HCF = q
Otherwise set p = q and q = r and repeat the above process.
Here we have p = 1073 and q = 551.
$1073=551\times 1+522$
Here r = 522.
So we set p = 551 and q = 522
Applying Euclid's Division lemma, we get
$551=522\times 1+29$
Here r = 29.
So we set p = 522 and q = 29.
Applying Euclid's Division lemma, we get
$522=29\times 18+0$
Since r = 0, we have HCF(551,1073) = 29.
Hence we have HCF(ab,bc) = 29.
But HCF(ab,bc)= b.
Hence we have b = 29.
Now $a=\dfrac{ab}{b}=\dfrac{551}{29}=19$ and $c=\dfrac{bc}{b}=\dfrac{1073}{29}=37$
Hence the numbers are 19,29 and 37.
Hence the sum of the numbers is 37+29+19=85.
Hence option [a] is correct.

Note: Alternatively, we have
If a and b are coprime and p is prime dividing a, then p does not divide b.
Also if a and b are any two numbers and p divides ab, then p divides a or p divides b.
Now we have $551=19\times 29$ and $1073=29\times 37$
Now we have 29 divides ab, then 29 divides a or 29 divides b.
Also, 29 divides bc, then 29 divides c or 29 divides b.
If 29 divides a, then we have 29 does not divide b and hence 29 divides c.
But since a and c are coprime, 29 cannot be a common factor.
Hence 29 divides b.
Also, every factor of b must divide both ab and bc, and since there is no common prime other than 29 among the numbers ab and bc, we have b = 29.
Hence a = 19 and c = 37, which is the same as obtained above.