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A given number is a multiple of $7$ . It is between $30$ and $60$ . It is also a factor of $70$ . What is the number?

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Solve system of equations: Complex word problems

Full solution

Q. A given number is a multiple of

7

. It is between

30

and

60

. It is also a factor of

70

. What is the number?

List Multiples of $7$ : List all multiples of $7$ between $30$ and $60$ . $\newline$ Multiples of $7$ between $30$ and $60$ are $35$ , $42$ , $49$ , and $56$ .
Identify Factors of $70$ : Identify which of these multiples are factors of $70$ . A factor of a number is a number that divides into it without leaving a remainder. We need to check which of the multiples of $7$ also divide $70$ without a remainder.
Check $35$ as Factor: Check if $35$ is a factor of $70$ . $\newline$ $70 \div 35 = 2$ , which means $35$ is a factor of $70$ .
Check $42$ as Factor: Check if $42$ is a factor of $70$ . $\newline$ $70 \div 42$ does not result in a whole number, so $42$ is not a factor of $70$ .
Check $49$ as Factor: Check if $49$ is a factor of $70$ . $\newline$ $70 \div 49$ does not result in a whole number, so $49$ is not a factor of $70$ .
Check $56$ as Factor: Check if $56$ is a factor of $70$ . $\newline$ $70 \div 56$ does not result in a whole number, so $56$ is not a factor of $70$ .
Conclude Correct Number: Conclude the number that meets all the criteria. $\newline$ Since $35$ is the only number that is a multiple of $7$ , between $30$ and $60$ , and also a factor of $70$ , it is the number we are looking for.

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