Question

1. a positive whole number less than 100 has remainder 2 when it is divided by 3, remainder 3 when it is divided by 4 and remainder 4 when it is divided by 5. what is its remainder when it is divided by 7?

Explanation:

Translate the problem into modular equations

Using the Modular Arithmetic knowledge point

$$
LATEXBLOCK0
$$

Analyze the pattern of the remainders

We observe a constant difference between the divisor and the remainder for each condition:

$$
LATEXBLOCK1
$$

This means that adding \(1\) to our number \(n\) makes it exactly divisible by \(3\), \(4\), and \(5\):

$$ n + 1 \equiv 0 \pmod 3, \quad n + 1 \equiv 0 \pmod 4, \quad n + 1 \equiv 0 \pmod 5 $$

Find the least common multiple

Using the Least Common Multiple knowledge point

$$
LATEXBLOCK2
$$

Determine the unique value of n

Since \(n\) is a positive whole number less than \(100\):

$$
LATEXBLOCK3
$$

Thus, the unique positive integer satisfying the conditions is \(n = 59\).

Calculate the remainder when divided by 7

We divide \(59\) by \(7\) to find the final remainder:

$$
LATEXBLOCK4
$$

The remainder is \(3\).

Answer:

3