Try this beautiful problem from Algebra based on multiplication and divisibility of two given numbers.
Multiplication and Divisibility- AMC 8, 2014
The 7-digit numbers 74A52B1 ana 326AB4C are each multiples of 3.which of the following could be the value of c ?
1
2
3
Key Concepts
Algebra
Division algorithm
Integer
Check the Answer
Answer:1
AMC-8, 2014 problem 21
Challenges and Thrills of Pre College Mathematics
Try with Hints
Use the rules of Divisibility ........
Can you now finish the problem ..........
If both numbers are divisible by 3 then the sum of their digits has to be divisible by 3......
can you finish the problem........
Since both numbers are divisible by 3, the sum of their digits has to be divisible by three. 7 + 4 + 5 + 2 + 1 = 19. In order to be a multiple of 3, A + B has to be either 2 or 5 or 8... and so on. We add up the numerical digits in the second number; 3 + 2 + 6 + 4 = 15. We then add two of the selected values, 5 to 15, to get 20. We then see that C = 1, 4 or 7, 10... and so on, otherwise the number will not be divisible by three. We then add 8 to 15, to get 23, which shows us that C = 1 or 4 or 7... and so on. In order to be a multiple of three, we select a few of the common numbers we got from both these equations, which could be 1, 4, and 7. However, in the answer choices, there is no 7 or 4 or anything greater than 7, but there is a 1. so the answer is 1
The least common multiple of a and b is 12 .and the lest common multiple of b and c is 15.what is the least possible value of the least common multiple of a and c?
30
60
20
Key Concepts
Algebra
Division algorithm
Integer
Check the Answer
Answer:20
AMC-8, 2016 problem 20
Challenges and Thrills of Pre College Mathematics
Try with Hints
Find greatest common factors
Can you now finish the problem ..........
Find Least common multiple....
can you finish the problem........
we wish to find possible values of a,b and c .By finding the greatest common factor 12 and 15, algebrically ,it's some multiple of b and from looking at the numbers ,we are sure that it is 3.Moving on to a and c ,in order to minimize them,we wish to find the least such that the LCM of a and 3 is 12,$\to 4$.similarly with 3 and c,we obtain 5.the LCM of 4 and 5 is 20 .
