Finding the Least Number with Specific Remainders

The problem asks us to find the smallest positive integer that satisfies two conditions:

• When divided by 12, 18, 24, and 30, it leaves a remainder of 4 in each case.
• When divided by 7, it leaves no remainder (i.e., it is exactly divisible by 7).

Step 1: Understanding the First Condition

A number that leaves the same remainder when divided by several different numbers can be expressed in a particular form. If a number leaves a remainder of 4 when divided by 12, 18, 24, and 30, it means that if we subtract 4 from this number, the result will be perfectly divisible by 12, 18, 24, and 30.

So, the number must be of the form \(\text{LCM}(12, 18, 24, 30) \times k + 4\), where \(k\) is a non-negative integer.

Step 2: Calculate the Least Common Multiple (LCM)

We need to find the LCM of 12, 18, 24, and 30. We can do this by finding the prime factorization of each number:

• \(12 = 2^2 \times 3^1\)
• \(18 = 2^1 \times 3^2\)
• \(24 = 2^3 \times 3^1\)
• \(30 = 2^1 \times 3^1 \times 5^1\)

The LCM is found by taking the highest power of each prime factor that appears in any of the factorizations:

\(\text{LCM}(12, 18, 24, 30) = 2^{\text{max}(2, 1, 3, 1)} \times 3^{\text{max}(1, 2, 1, 1)} \times 5^{\text{max}(0, 0, 0, 1)}\)

\(\text{LCM}(12, 18, 24, 30) = 2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 = 72 \times 5 = 360\)

Step 3: Formulate the General Expression for the Number

Based on the first condition, the number must be of the form \(360k + 4\), where \(k\) is an integer (\(k \ge 0\) since we are looking for a positive number). The possible numbers are \(4, 364, 724, 1084, \dots\).

Step 4: Apply the Second Condition

The second condition states that the number must be divisible by 7. So, when we divide \(360k + 4\) by 7, the remainder must be 0.

We can write this as: \((360k + 4) \equiv 0 \pmod{7}\)

Let's find the remainder of 360 when divided by 7:

\(360 \div 7 = 51\) with a remainder of \(3\). So, \(360 \equiv 3 \pmod{7}\).

Substitute this into the congruence:

\((3k + 4) \equiv 0 \pmod{7}\)

We need to find the smallest non-negative integer value of \(k\) that satisfies this condition.

Step 5: Find the Value of k

We test values of \(k\) starting from 0:

• If \(k=0\), \(3(0) + 4 = 4\). \(4 \pmod{7} = 4\). (Not divisible by 7)
• If \(k=1\), \(3(1) + 4 = 7\). \(7 \pmod{7} = 0\). (Divisible by 7)

The smallest non-negative value of \(k\) that satisfies the condition is \(k=1\).

Step 6: Calculate the Least Number

Substitute \(k=1\) into the general form \(360k + 4\):

Least number = \(360(1) + 4 = 360 + 4 = 364\)

Step 7: Verification

Let's check if 364 satisfies both conditions:

• Divided by 12: \(364 \div 12 = 30\) with remainder \(4\). (\(12 \times 30 = 360\))
• Divided by 18: \(364 \div 18 = 20\) with remainder \(4\). (\(18 \times 20 = 360\))
• Divided by 24: \(364 \div 24 = 15\) with remainder \(4\). (\(24 \times 15 = 360\))
• Divided by 30: \(364 \div 30 = 12\) with remainder \(4\). (\(30 \times 12 = 360\))
• Divided by 7: \(364 \div 7 = 52\) with remainder \(0\). (\(7 \times 52 = 364\))

Both conditions are met. The least number is 364.

Answer

The least number which when divided by 12, 18, 24 and 30 leaves 4 as remainder in each case, but when divided by 7 leaves no remainder is 364.

Revision Table: Least Number Problem

Additional Information: Number Theory Concepts

This problem combines the concepts of LCM and divisibility. Problems of this type often involve finding a number that satisfies multiple conditions related to remainders upon division.

• Relationship between LCM and GCD: For any two positive integers \(a\) and \(b\), \(\text{LCM}(a, b) \times \text{GCD}(a, b) = a \times b\). While not directly used in this specific problem with multiple numbers, the relationship is fundamental in number theory.
• Chinese Remainder Theorem: For more complex problems involving different remainders for different divisors, the Chinese Remainder Theorem (CRT) is a powerful tool. However, for problems where the remainder is the same for all divisors (like the first condition here), the LCM method is simpler. The second condition being simple divisibility by 7 makes the modular arithmetic approach effective.
• General form of numbers with specific remainders: A number that leaves a remainder \(r\) when divided by numbers \(d_1, d_2, \dots, d_n\) is of the form \(\text{LCM}(d_1, d_2, \dots, d_n) \times k + r\), provided \(r < d_i\) for all \(i\).