Dividing a Number by 2, 3, and 5: Understanding the Quotients and Remainders

When working with division, understanding the behavior of numbers under different divisors can provide fascinating insights. Let's explore a unique scenario where a number, when successively divided by 2, 3, and 5, yields specific quotients. This exploration will also help us determine what the quotient and remainder would be when dividing by 30.

Understanding the Given Problem

Consider the following conditions:

When a number is divided by 2, the quotient is 1. When the number is divided by 3, the quotient is 2. When the number is divided by 5, the quotient is 5.

Let's denote the number by N, the first quotient by Q1, the second quotient by Q2, and the third quotient by Q3. We can write the following equations:

N 2Q1 - 1 N 3Q2 - 2 N 5Q3 - 3

Solving the Equations

First, let's solve the equation for division by 2 and 3:

N 2Q1 - 1

From the first equation, we can say:

N ≡ -1 (mod 2) (1)

N ≡ -1 (mod 3) (2)

Combining (1) and (2), we get:

N ≡ -1 (mod 6) (3)

Thus, N -1 6k, where k is an integer.

N -6k - 1

Now, let's solve for division by 5:

N 5Q3 - 3

N ≡ 3 (mod 5) (4)

We have:

-6k - 1 ≡ 3 (mod 5)

-6k ≡ 4 (mod 5)

-6k ≡ -1 (mod 5)

Since -1 ≡ 4 (mod 5), we get:

6k ≡ 1 (mod 5)

From the above, we can solve for k:

k ≡ 1 (mod 5)

k 5m 1, where m is an integer.

Substituting back into the expression for N, we get:

N -6(5m 1) - 1

N -30m - 6 - 1

N -30m - 7

So, the remainder when N is divided by 30 is 23.

N ≡ 23 (mod 30)

Thus, the remainder when the number is divided by 30 is 23 (boxed 23).

Another Approach

Consider a number N and its quotients Q1, Q2, and Q3 when divided by 2, 3, and 5 respectively. Let's solve the problem step-by-step:

1. N 2Q1 - 1

2. N 3Q2 - 2

3. N 5Q3 - 3

From (1) and (2), we get:

N ≡ -1 (mod 6)

And from (2) and (3), we get:

N ≡ -1 (mod 30)

Thus, the number N can be expressed as:

N 30m 23

When N is divided by 30, the remainder is 23.

Case Analysis for Specific Numbers

To further clarify, let's consider specific values for N and check the conditions. If the number is 23, then:

23 2(11) - 1

23 3(8) - 2

23 5(4) - 3

Thus, 23 satisfies all the conditions given. When 23 is divided by 30, the remainder is 23.

Therefore, the remainder when the number is divided by 30 is 23.