What is the greatest positive integer that divides 554, 714 and 213 leaving the remainder 43, 57 and 67, respectively

What is the greatest positive integer that divides 554, 714 and 213 leaving the remainder 43, 57 and 67, respectively?

a)            70

b)            73

c)            77

d)            75

Sol:

Given: 

Integers 554, 714 and 213  

Concept Used:  HCF = Highest Common Factor 

Calculation: 

⇒ According to the question, 

⇒ If the integer leaves remainder 43, on dividing 554, then it divides 511 exactly 

⇒ If the integer leaves remainder 57, on dividing 714, then it divides 657 exactly 

⇒ If the integer leaves remainder 67, on dividing 213, then it divides 146 exactly 

⇒ On prime factorizing 511, 657 and 146, we have  

⇒ 511 = 7 × 73 

⇒ 657 = 3 × 3 × 73 

⇒ 146 = 2 × 73 

⇒ HCF of 511, 657 and 146 = 73 

Therefore, the greatest positive integer that divides 554, 714 and 213 leaving the remainder 43, 57 and 67, respectively is 73.