Remainders Affect My Mod

Can anyone guide on how I can solve the following problem without using trial-and-error:

A number when divided by 5 leaves a remainder of 2 and when divided by 7, leaves a remainder of 4. What is the remainder when the same number is divided by $5 \times 7$ ?

#NumberTheory

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Chinese Remainder Theorem.

Pi Han Goh - 4 years, 7 months ago

It is 32.

Sarthak Rout - 4 years, 5 months ago

I got that answer by trial-and-error. But, may you please explain the mathematical way to solve it?

Nilabha Saha - 4 years, 5 months ago

You see, let us take the no. to be X. And € represent congruence. X € 2 (mod 5) and X € 4 (mod 7). X € 2 (mod 5)= X € 2-5 (mod 5)=X € -3 (mod 5) & similarly, X € 4 (mod 7)= X € 4-7 (mod 7)=X € -3 (mod 7). Now, X € -3 (mod 5×7)= X € -3 (mod 35)=X € -3 +35 (mod 35)=X € 32 (mod 35). So, the remainder left when divided by 35 is 32.

Sarthak Rout - 4 years, 5 months ago

@Sarthak Rout – I see. It all makes sense now. Thank you again for helping me understand the mathematics behind solving this problem.

Nilabha Saha - 4 years, 5 months ago

@Nilabha Saha – You are welcome.

Sarthak Rout - 4 years, 5 months ago

You see let the number be X. Then X is of the form 5a+2 or 5b-3. Similarly, X is of tge form, 7c+4 or 7d-3. So, when divided by 35 , we can say it will leave remainder -3 which is common both cases. So, X = 35e -3 which corresponds to a remainder of 35-3 = 32. You can do it in congruent modulo concept too.

Sarthak Rout - 4 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$