Understanding the Problem: Finding the Smallest Natural Number

The question asks for the smallest natural number that leaves a remainder of 4 when divided by 5, 6, 7, or 8. A natural number is a positive integer (1, 2, 3, ...).

This means if we call the number $N$, then:

• $N \div 5$ gives a remainder of 4.
• $N \div 6$ gives a remainder of 4.
• $N \div 7$ gives a remainder of 4.
• $N \div 8$ gives a remainder of 4.

If a number leaves the same remainder $r$ when divided by several different numbers $d_1, d_2, \dots, d_k$, then the number can be expressed in the form $M \cdot L + r$, where $L$ is the Least Common Multiple (LCM) of $d_1, d_2, \dots, d_k$, and $M$ is a non-negative integer (0, 1, 2, ...).

In this problem, the divisors are 5, 6, 7, and 8, and the remainder is 4.

Calculating the Least Common Multiple (LCM)

First, we need to find the LCM of the divisors: 5, 6, 7, and 8. To find the LCM, we can use the prime factorization method.

• Prime factorization of 5 is 5.
• Prime factorization of 6 is 2 × 3.
• Prime factorization of 7 is 7.
• Prime factorization of 8 is 2 × 2 × 2, which is $2^3$.

The LCM is found by taking the highest power of all prime factors that appear in any of the numbers:

• Highest power of 2 is $2^3$ (from 8).
• Highest power of 3 is $3^1$ (from 6).
• Highest power of 5 is $5^1$ (from 5).
• Highest power of 7 is $7^1$ (from 7).

So, LCM(5, 6, 7, 8) = $2^3 \times 3^1 \times 5^1 \times 7^1 = 8 \times 3 \times 5 \times 7$.

Calculation:

$8 \times 3 = 24$

$24 \times 5 = 120$

$120 \times 7 = 840$

The LCM of 5, 6, 7, and 8 is 840.

Finding the Smallest Natural Number

The numbers that leave a remainder of 4 when divided by 5, 6, 7, or 8 are of the form:

Number = (LCM of 5, 6, 7, 8) × k + Remainder

Number = $840 \times k + 4$, where $k$ is a non-negative integer ($k \ge 0$).

We are looking for the smallest natural number of this form.

• If $k=0$: Number = $840 \times 0 + 4 = 0 + 4 = 4$. The number 4 is a natural number. Let's check if 4 leaves a remainder of 4 when divided by 5, 6, 7, or 8.
  • $4 \div 5$: quotient 0, remainder 4 (Correct)
  • $4 \div 6$: quotient 0, remainder 4 (Correct)
  • $4 \div 7$: quotient 0, remainder 4 (Correct)
  • $4 \div 8$: quotient 0, remainder 4 (Correct)

  So, 4 satisfies the condition. However, looking at the options provided (1264, 844, 424, 214), 4 is not listed.

• If $k=1$: Number = $840 \times 1 + 4 = 840 + 4 = 844$. The number 844 is a natural number. Let's check if 844 leaves a remainder of 4 when divided by 5, 6, 7, or 8.
  • $844 \div 5 = (840 + 4) \div 5$. Since 840 is divisible by 5, the remainder is 4. (Correct)
  • $844 \div 6 = (840 + 4) \div 6$. Since 840 is divisible by 6, the remainder is 4. (Correct)
  • $844 \div 7 = (840 + 4) \div 7$. Since 840 is divisible by 7, the remainder is 4. (Correct)
  • $844 \div 8 = (840 + 4) \div 8$. Since 840 is divisible by 8, the remainder is 4. (Correct)

  The number 844 satisfies the condition and is listed in the options.

The numbers satisfying the condition are 4, 844, $840 \times 2 + 4 = 1684$, and so on. The smallest among these is 4. However, given the options, the smallest number from the options that fits this form is 844.

Therefore, based on the options provided, the smallest number is 844.

Verification with Options

Let's quickly check the other options:

• 1264: $1264 = 840 \times 1 + 424$. Does not fit the form $840k + 4$.
• 844: $844 = 840 \times 1 + 4$. Fits the form. Leaves remainder 4 when divided by 5, 6, 7, or 8.
• 424: $424 = 840 \times 0 + 424$. Does not fit the form $840k + 4$. $424 \div 5 = 84$ rem 4 (Correct), $424 \div 6 = 70$ rem 4 (Correct), $424 \div 7 = 60$ rem 4 (Correct), $424 \div 8 = 53$ rem 0 (Incorrect - remainder should be 4).
• 214: $214 = 840 \times 0 + 214$. Does not fit the form $840k + 4$. $214 \div 8 = 26$ rem 6 (Incorrect - remainder should be 4).

Only 844 from the options satisfies all the conditions.

Revision Table: Number Theory Concepts

• Natural Numbers: The set of positive integers $\{1, 2, 3, \dots\}$. Sometimes includes 0, but commonly starts from 1.
• Remainder: In division $a \div b$, if $a = qb + r$ where $0 \le r < |b|$, then $r$ is the remainder.
• Least Common Multiple (LCM): The smallest positive integer that is a multiple of two or more given integers.

Additional Information: Numbers with Same Remainder

Problems involving finding a number that leaves the same remainder when divided by multiple numbers are common in number theory. The general approach is to find the LCM of the divisors and add the common remainder. Any number of the form $k \times \text{LCM} + \text{remainder}$ will satisfy the condition, where $k$ is a non-negative integer.

If the remainder is 0, the number is simply a multiple of the LCM.

If the number is asked to be within a certain range, you test values of $k$ (starting from 0 or 1, depending on whether 0 is included or if a 'natural' number implies positive) until you find a number in that range.

In this specific problem, the numbers are of the form $840k + 4$. The sequence of such numbers is $4, 844, 1684, 2524, \dots$. The smallest natural number in this sequence is 4. However, given the multiple-choice options, the question likely intends to ask for the smallest number *among the options* that satisfies the criteria, or perhaps the smallest positive integer of this form, which is 844 (obtained when $k=1$). Since 844 is an option and satisfies the condition, it is the intended answer.