Solved on Jan 24, 2024

Find the least possible number that when divided by 6 leaves a remainder of 2, and when divided by 7 leaves a remainder of 3. Determine if Daniel's claim that the least number is 38 is correct.

STEP 1

Assumptions
1. When the number is divided by 6, the remainder is 2.
2. When the same number is divided by 7, the remainder is 3.
3. We are looking for the least possible number that satisfies both conditions.

STEP 2

We can express the conditions using the following congruences:
$\begin{align} n &\equiv 2 \pmod{6} \\ n &\equiv 3 \pmod{7} \end{align}$

STEP 3

The first congruence tells us that the number can be expressed as:
$n = 6k + 2$
where $k$ is an integer.

STEP 4

The second congruence tells us that the number can also be expressed as:
$n = 7m + 3$
where $m$ is an integer.

STEP 5

We need to find the smallest number $n$ that satisfies both equations. This means finding the smallest common value for $6k + 2$ and $7m + 3$ .

STEP 6

We can start by listing multiples of 6 and adding 2 to find numbers that satisfy the first condition:
$8, 14, 20, 26, 32, 38, \ldots$

STEP 7

Similarly, we can list multiples of 7 and add 3 to find numbers that satisfy the second condition:
$10, 17, 24, 31, 38, \ldots$

STEP 8

We look for the smallest number that appears in both lists. From the lists, we see that 38 is the smallest number that satisfies both conditions.

STEP 9

Therefore, we agree with Daniel that the least possible number that satisfies both conditions is 38.
The least possible number that satisfies both conditions is 38.

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Find the least possible number that when divided by 6 leaves a remainder of 2, and when divided by 7 leaves a remainder of 3. Determine if Daniel's claim that the least number is 38 is correct.

STEP 1

Assumptions
1. When the number is divided by 6, the remainder is 2.
2. When the same number is divided by 7, the remainder is 3.
3. We are looking for the least possible number that satisfies both conditions.

STEP 2

We can express the conditions using the following congruences:
$\begin{align} n &\equiv 2 \pmod{6} \\ n &\equiv 3 \pmod{7} \end{align}$

STEP 3

The first congruence tells us that the number can be expressed as:
$n = 6k + 2$
where $k$ is an integer.

STEP 4

The second congruence tells us that the number can also be expressed as:
$n = 7m + 3$
where $m$ is an integer.

STEP 5

We need to find the smallest number $n$ that satisfies both equations. This means finding the smallest common value for $6k + 2$ and $7m + 3$ .

STEP 6

We can start by listing multiples of 6 and adding 2 to find numbers that satisfy the first condition:
$8, 14, 20, 26, 32, 38, \ldots$

STEP 7

Similarly, we can list multiples of 7 and add 3 to find numbers that satisfy the second condition:
$10, 17, 24, 31, 38, \ldots$

STEP 8

We look for the smallest number that appears in both lists. From the lists, we see that 38 is the smallest number that satisfies both conditions.

STEP 9

Therefore, we agree with Daniel that the least possible number that satisfies both conditions is 38.
The least possible number that satisfies both conditions is 38.

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Find the least possible number that when divided by 6 leaves a remainder of 2, and when divided by 7 leaves a remainder of 3. Determine if Daniel's claim that the least number is 38 is correct.

STEP 1

Assumptions1. When the number is divided by 6, the remainder is 2.2. When the same number is divided by 7, the remainder is 3.3. We are looking for the least possible number that satisfies both conditions.

STEP 2

We can express the conditions using the following congruences:n≡2(mod6)n≡3(mod7) \begin{align*} n &\equiv 2 \pmod{6} \\ n &\equiv 3 \pmod{7} \end{align*} nn​≡2(mod6)≡3(mod7)​

STEP 3

The first congruence tells us that the number can be expressed as:n=6k+2n = 6k + 2n=6k+2where kkk is an integer.

STEP 4

The second congruence tells us that the number can also be expressed as:n=7m+3n = 7m + 3n=7m+3where mmm is an integer.

STEP 5

We need to find the smallest number nnn that satisfies both equations. This means finding the smallest common value for 6k+26k + 26k+2 and 7m+37m + 37m+3.

STEP 6

We can start by listing multiples of 6 and adding 2 to find numbers that satisfy the first condition:8,14,20,26,32,38,…8, 14, 20, 26, 32, 38, \ldots8,14,20,26,32,38,…

STEP 7

Similarly, we can list multiples of 7 and add 3 to find numbers that satisfy the second condition:10,17,24,31,38,…10, 17, 24, 31, 38, \ldots10,17,24,31,38,…

STEP 8

We look for the smallest number that appears in both lists. From the lists, we see that 38 is the smallest number that satisfies both conditions.

STEP 9

Therefore, we agree with Daniel that the least possible number that satisfies both conditions is 38.The least possible number that satisfies both conditions is 38.

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Assumptions
1. When the number is divided by 6, the remainder is 2.
2. When the same number is divided by 7, the remainder is 3.
3. We are looking for the least possible number that satisfies both conditions.

We can express the conditions using the following congruences:
$\begin{align} n &\equiv 2 \pmod{6} \\ n &\equiv 3 \pmod{7} \end{align}$

The first congruence tells us that the number can be expressed as:
$n = 6k + 2$
where $k$ is an integer.

The second congruence tells us that the number can also be expressed as:
$n = 7m + 3$
where $m$ is an integer.

We need to find the smallest number $n$ that satisfies both equations. This means finding the smallest common value for $6k + 2$ and $7m + 3$ .

We can start by listing multiples of 6 and adding 2 to find numbers that satisfy the first condition:
$8, 14, 20, 26, 32, 38, \ldots$

Similarly, we can list multiples of 7 and add 3 to find numbers that satisfy the second condition:
$10, 17, 24, 31, 38, \ldots$

Therefore, we agree with Daniel that the least possible number that satisfies both conditions is 38.
The least possible number that satisfies both conditions is 38.