Prove that the expression \(9 \times 7 + 3 \times 12\) is divisible by 90, and that \(2 \times 5 \times 9 + 5 \times 17\) is divisible by 30.

Mathematics Questions

Prove that the expression (9 times 7 + 3 times 12) is divisible by 90, and that (2 times 5 times 9 + 5 times 17) is divisible by 30.

Short Answer

The process involves proving the divisibility of two expressions, 9 7 + 3 12 by 90 and 2 5 9 + 5 17 by 30. Through simplification and factoring, it is demonstrated that both expressions can be confirmed as divisible by their respective numbers.

Step-by-Step Solution

Step 1: Understand the Numbers and Their Divisibility

The first step is to identify the numbers involved in the expressions we are proving to be divisible. Here we have two expressions: 9 7 + 3 12 and 2 5 9 + 5 17. We need to check their divisibility by 90 and 30 respectively. Understanding the structure of these numbers will help in manipulating them correctly to reach a solution.

Step 2: Simplify the Expressions

Next, we will simplify each expression to facilitate the proof of divisibility. For 9 7 + 3 12, this can be transformed into:

(3 2) 7 + 3 12
3 14 + 3 12
3 12 (3 2 + 1) = 3 12 * 10
This results in 3 10 * 90

Thus, it confirms that 9 7 + 3 12 is divisible by 90.

Step 3: Conclude with Second Expression

For the second expression, 2 5 9 + 5 17, we will apply a similar simplification process:

Start with: 2 5 9 + 5 17
Rewrite it as: 5 18 + 5 17
Factoring gives us: 5 17 (5 + 1) = 5 16 ‚àó 30

This shows that 2 5 9 + 5 17 is also divisible by 30. Thus, both expressions have been proven to be divisible as required.

Prove that the expression (9 times 7 + 3 times 12) is divisible by 90, and that (2 times 5 times 9 + 5 times 17) is divisible by 30.