What times what equals 252?

Understand the Problem

The question is asking for pairs of numbers that, when multiplied together, result in 252. This implies finding all factor pairs of 252 or solving the equation x * y = 252 for positive integers x and y.

Answer

The pairs of numbers that multiply to 252 are: $(1, 252), (2, 126), (3, 84), (4, 63), (6, 42), (7, 36), (9, 28), (12, 21), (14, 18)$

Steps to Solve

Find the factors of 252

To find the pairs of numbers that multiply to 252, we first need to determine its factors. A factor is a number that divides another without leaving a remainder. We can start dividing 252 by integers beginning from 1 up to the square root of 252 (approximately 15.87).

List the pairs of factors

For every factor ( f ) found, we can find its corresponding pair ( g ) using the formula ( g = \frac{252}{f} ).

Calculate the factors systematically

We systematically check for every integer from 1 to 15:

( 1 \times 252 = 252 )
( 2 \times 126 = 252 )
( 3 \times 84 = 252 )
( 4 \times 63 = 252 )
( 6 \times 42 = 252 )
( 7 \times 36 = 252 )
( 9 \times 28 = 252 )
( 12 \times 21 = 252 )
( 14 \times 18 = 252 )

List all unique pairs

After finding all factor pairs, we list them, ensuring not to repeat pairs (e.g., ( 18 \times 14 ) is the same as ( 14 \times 18 )).

The complete list of pairs:

( (1, 252) )
( (2, 126) )
( (3, 84) )
( (4, 63) )
( (6, 42) )
( (7, 36) )
( (9, 28) )
( (12, 21) )
( (14, 18) )

The pairs of numbers that multiply to 252 are:

$ (1, 252), (2, 126), (3, 84), (4, 63), (6, 42), (7, 36), (9, 28), (12, 21), (14, 18) $

More Information

The number 252 can be decomposed into various pairs, and these pairs signify the different combinations of whole numbers that can yield the value of 252 when multiplied. Each factorization represents key relationships in multiplication and division.

Tips

Forgetting to check all numbers up to 15. Make sure to check each integer up to the square root to find all possible pairs.
Repeating pairs in the final list. Ensure each pair is unique to maintain clarity.

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