Finding the Smallest Number with a Modulo Remainder of 13

Understanding how to find the smallest number that leaves a specific remainder when divided by multiple divisors is a fundamental concept in number theory. This article delves into the method to find the smallest number which, when divided by 25, 40, or 56, leaves a remainder of 13 in each case. We'll explore the mathematical principles, including congruences and the least common multiple (LCM), and apply them to solve this problem.

Understanding the Problem

Given that we need to find the smallest number ( x ) which gives a remainder of 13 when divided by 25, 40, and 56, we can start by setting up the following congruences:

( x equiv 13 (text{mod} 25) )

( x equiv 13 (text{mod} 40) )

( x equiv 13 (text{mod} 56) )

Solving the Problem Using Congruences

To solve this, let's introduce a new variable ( y ) such that:

( y x - 13 )

Therefore, ( y ) must be divisible by 25, 40, and 56. Our task now is to find the least common multiple (LCM) of these three numbers.

Calculating the Lowest Common Multiple (LCM)

First, we factorize the numbers:

25 ( 5^2 )

40 ( 2^3 times 5^1 )

56 ( 2^3 times 7^1 )

Next, we find the highest power of each prime number that appears in the factorization of these numbers:

For ( 2 ), the highest power is ( 2^3 ) from both 40 and 56.

For ( 5 ), the highest power is ( 5^2 ) from 25.

For ( 7 ), the highest power is ( 7^1 ) from 56.

Thus, the LCM is calculated as:

LCM ( 2^3 times 5^2 times 7^1 )

Calculating step by step:

8 × 25 200

200 × 7 1400

Hence, the LCM of 25, 40, and 56 is 1400.

Expressing ( x ) in Terms of ( y )

Since ( y x - 13 ) must be a multiple of 1400, we express ( y ) as:

( y 1400k quad text{for some integer } k )

Thus, we can find ( x ) as follows:

( x 1400k 13 )

Identifying the Smallest Positive Number

To find the smallest positive ( x ), we set ( k 1 ):

( x 1400 times 1 13 1413 )

Therefore, the smallest number which when divided by 25, 40, or 56 leaves a remainder of 13 is 1413.

Alternative Method for Finding the LCM

Alternatively, another method to find the LCM of 25, 40, and 56 involves identifying the highest powers of the primes that appear in the factorization:

25 ( 5^2 )

40 ( 2^3 times 5 )

56 ( 2^3 times 7 )

Thus, LCM ( 2^3 times 5^2 times 7 1400 )

Adding the remainder (13) gives the smallest number as 1413.

Conclusion

The process of finding the smallest number that leaves a specific remainder when divided by multiple divisors involves systematically using congruences and the least common multiple. By applying these principles, we can solve problems involving modular arithmetic in a structured manner.