Question 6
Q6: For two positive integers A & B, what is the highest number that divides completely the product of integers from 1 to A and 1 to B such that B = A + 29.
- 1
- Product of all integers from 1 to A
- Product of all integers from 1 to B
- 29*A
- Can’t be determined
Correct Answer
B
Solution
Given
We are given two positive integers A & B. We are asked to find the highest number which divides product of all integers from 1 to A product of all integers from 1 to B. We are also given that B = A +29
Approach
The highest number which divides two numbers is the GCD of the numbers. So the question is asking us to find the GCD of two numbers.
The first number is the product of all integers from 1 to A. This product is known as A Factorial and is represented A!
We can write A! = 1* 2* 3……A.
Similarly product of all integers from 1 to B is known as B!
We can write B! = 1* 2* 3…………..B.
So, we are asked to find the GCD (A! , B!). Since, we are given that B = A + 29 it would imply that B > A
Let’s use our understanding of GCD to find out the GCD (A! , B!)
Working Out
Since B > A, we can write B! = 1* 2* 3…..A * (A+1)*(A+2)……B.
We know that 1*2*3….. A = A!.
So, B! = A! * (A+1)*(A+2)…B.
We see here that A! is a factor of both A! and B!. We also know that GCD of a set of numbers can’t be greater than the magnitude of the smallest number which in this case is A!
Hence, we can say that A! is the highest number which divides both A! and B!. Therefore GCD(A! , B!) = A! i.e. product of all integers from 1 to A
Answer: Option (B)
