The multiples of 12 and of 21 are written in order. What is the smallest positive difference between a multiple of 12 and a multiple of 21?

Guest Dec 2, 2017

#1**+2 **

Well 12 x 21 = 252 so the smallest possible difference is ZERO.

If you want the smallest possible POSITIVE difference, let's write out the multiples

12 24 36 48 60 72 84 96 108 120 132 144 156 168

21 42 63 84 105 126 147 168 This repeats every 4th multiple of 21 (or 7th multiple of 12)

The LEAST POSITIVE difference is 3 it appears .

ElectricPavlov
Dec 2, 2017

#1**+2 **

Best Answer

Well 12 x 21 = 252 so the smallest possible difference is ZERO.

If you want the smallest possible POSITIVE difference, let's write out the multiples

12 24 36 48 60 72 84 96 108 120 132 144 156 168

21 42 63 84 105 126 147 168 This repeats every 4th multiple of 21 (or 7th multiple of 12)

The LEAST POSITIVE difference is 3 it appears .

ElectricPavlov
Dec 2, 2017

#2**+1 **

**The multiples of 12 and of 21 are written in order. **

**What is the smallest positive difference between a multiple of 12 and a multiple of 21?**

The difference is \(n\cdot 21 -m \cdot 12 = d \) with \( n,m,d \in Z\) has a solution,

if the \(gcd(21,12)\) is a divider of \(d\). Or \(gcd(21,12) | d\).

The greatest common divisor \(gcd(21,12) = 3.\)

The difference is \(n\cdot gcd(21,12)\)

so the difference is a **multiple of 3**

The differences are: \(0, 3, 6, 9, 12, \ldots\)

The smallest positive difference is **3**.

heureka
Dec 4, 2017