15.A certain number when successively divided by 3 and 5 leaves reminder 1 and 2. what is the reminder, if the same number be divided by 15?Options58746

1. Break Down the Problem

We are given a number $n$ that leaves a remainder of 1 when divided by 3 and a remainder of 2 when divided by 5. We are required to find the remainder of $n$ when it is divided by 15.

2. Relevant Concepts

The conditions given can be expressed in mathematical terms:

• $n \equiv 1 \pmod{3}$
• $n \equiv 2 \pmod{5}$

We will use the method of solving simultaneous congruences.

3. Analysis and Detail

To solve the system of congruences, we can express $n$ in terms of one of the modular equations.

From $n \equiv 2 \pmod{5}$:

• This means $n = 5k + 2$ for some integer $k$.

Substituting this into the first congruence:

• $5k + 2 \equiv 1 \pmod{3}$

Reducing $5k + 2$ modulo 3:

• $5 \equiv 2 \pmod{3}$, thus $5k \equiv 2k \pmod{3}$.
• Therefore, the equation becomes $2k + 2 \equiv 1 \pmod{3}$, which simplifies to $2k \equiv -1 \equiv 2 \pmod{3}$.

Now, solving for $k$:

• Multiplying both sides by the modular inverse of 2 mod 3, which is also 2, we get $k \equiv 4 \equiv 1 \pmod{3}$.
• This implies $k = 3m + 1$ for some integer $m$.

Substituting back to find $n$:

• $n = 5(3m + 1) + 2 = 15m + 5 + 2 = 15m + 7$.

Thus, we have $n \equiv 7 \pmod{15}$.

4. Verify and Summarize

Now, to find the remainder when $n$ is divided by 15, we determine $n \mod 15$:

• Since $n = 15m + 7$, it follows that $n \equiv 7 \mod{15}$.

Final Answer

The remainder when the number is divided by 15 is 7.