A number divides 12288, 28200 and 44333 so as to leave the same remainder in each case. What is that number?

Question

Prepp Team · Accepted Answer

Finding the Number with the Same Remainder

The problem asks us to find a number that divides three given numbers, 12288, 28200, and 44333, leaving the same remainder in each case. Let the three numbers be \(a\), \(b\), and \(c\), and the unknown number be \(n\). If \(n\) divides \(a\), \(b\), and \(c\) with the same remainder \(r\), we can write this using the division algorithm:

\(a = n \times q_1 + r\)
\(b = n \times q_2 + r\)
\(c = n \times q_3 + r\)

where \(q_1\), \(q_2\), and \(q_3\) are the quotients.

If we take the difference between any two of these numbers, the remainder \(r\) will be eliminated:

\(b - a = (n \times q_2 + r) - (n \times q_1 + r) = n \times (q_2 - q_1)\)
\(c - b = (n \times q_3 + r) - (n \times q_2 + r) = n \times (q_3 - q_2)\)
\(c - a = (n \times q_3 + r) - (n \times q_1 + r) = n \times (q_3 - q_1)\)

This shows that the number \(n\) must be a divisor of the differences between the numbers. Therefore, \(n\) must be a common divisor of \((b-a)\), \((c-b)\), and \((c-a)\). The largest such number would be the Greatest Common Divisor (GCD) of these differences.

Let's calculate the differences between the given numbers:

Difference between 28200 and 12288: \(28200 - 12288 = 15912\)
Difference between 44333 and 28200: \(44333 - 28200 = 16133\)
Difference between 44333 and 12288: \(44333 - 12288 = 32045\)

The number we are looking for is a common divisor of 15912, 16133, and 32045. Let's find the GCD of these differences. We can start by finding the GCD of two of the differences, say 15912 and 16133, using the Euclidean algorithm.

Applying the Euclidean Algorithm:

\(16133 = 15912 \times 1 + 221\)

\(15912 = 221 \times 72 + 0\)

The GCD of 15912 and 16133 is 221.

Now we need to check if 221 is also a divisor of the third difference, 32045.

\(32045 \div 221\)

\(32045 = 221 \times 145\)

Since 221 divides 32045 exactly, 221 is a common divisor of all three differences: 15912, 16133, and 32045. As 221 is the GCD of the first two differences and it divides the third, it is the GCD of all three differences. This makes 221 the largest possible number that satisfies the condition.

We can verify this by dividing the original numbers by 221:

\(12288 \div 221\): \(12288 = 221 \times 55 + 133\). The remainder is 133.
\(28200 \div 221\): \(28200 = 221 \times 127 + 133\). The remainder is 133.
\(44333 \div 221\): \(44333 = 221 \times 200 + 133\). The remainder is 133.

Indeed, when 12288, 28200, and 44333 are divided by 221, the remainder is 133 in all cases.

Checking the options, 221 is present and fits our finding. Let's quickly check other options to ensure they don't leave the same remainder:

Number	Divided by 272	Divided by 232	Divided by 120
12288	\(12288 = 272 \times 45 + 48\) (Rem 48)	\(12288 = 232 \times 53 + 12\) (Rem 12)	\(12288 = 120 \times 102 + 48\) (Rem 48)
28200	\(28200 = 272 \times 103 + 184\) (Rem 184)	\(28200 = 232 \times 121 + 88\) (Rem 88)	\(28200 = 120 \times 235 + 0\) (Rem 0)
44333	\(44333 = 272 \times 163 + 37\) (Rem 37)	\(44333 = 232 \times 191 + 201\) (Rem 201)	\(44333 = 120 \times 369 + 53\) (Rem 53)

As seen in the table, none of the other options (272, 232, 120) result in the same remainder for all three numbers.

Therefore, the number that divides 12288, 28200, and 44333 so as to leave the same remainder in each case is 221.

Revision Table: Same Remainder Problems

Concept	Explanation	Application
Same Remainder	If \(a \div n\) gives remainder \(r\), \(b \div n\) gives remainder \(r\), etc., then \(a-b\), \(b-c\), \(a-c\) are all divisible by \(n\).	Used to find the number \(n\) by looking at the differences between the given numbers.
Greatest Common Divisor (GCD)	The largest positive integer that divides two or more integers without leaving a remainder.	The required number is the GCD of the differences between the given numbers.
Euclidean Algorithm	An efficient method for computing the GCD of two integers.	Used to find the GCD of the differences (15912 and 16133).

Additional Information: Number Properties and GCD

This problem is a classic example of applying the concept of Greatest Common Divisor (GCD) in number theory. When a number divides two different numbers leaving the same remainder, say \(a = n q_1 + r\) and \(b = n q_2 + r\), subtracting them gives \(a - b = n(q_1 - q_2)\). This shows that the difference \((a-b)\) is divisible by \(n\). This principle extends to multiple numbers.

Finding the GCD of large numbers is made easy by the Euclidean algorithm, which involves repeated division. The algorithm relies on the property that the GCD of two numbers does not change if the larger number is replaced by its difference with the smaller number. This process continues until one of the numbers becomes zero, and the other number is the GCD.

Understanding divisibility rules and properties of remainders is fundamental in solving problems like this. The ability to set up equations based on the problem description and manipulate them (like subtracting to eliminate the remainder) is a key algebraic skill useful in number theory problems.