#### How many fractions of type ^{1a9b}/_{1x23y} can be simplified by 44?

^{1a9b}/

_{1x23y}

**Answer**

44 has to divide both *1a9b* and *1x23y*.

But 44 =4 * 11, therefore 4 and 11 must divide both *1a9b* and *1x23y*.

What are the divisibility rules for 4 and 11?

**“4 divides 1a9b”** means that

*b*= 2 or

*b*= 6 (last two digits of the number must be divisible by 4 for the given number to be divisible by 4).

For ** b = 2**, “11 divides

*1a92*” means that (1 –

*a*+ 9 – 2) = (8 –

*a*) is divisible by 11 (a number is

*divisible by 11*if the

*alternating sum*of the digits is

*divisible by 11*).

Since *a* is a natural number, (8 – *a*) is divisible by 11 only if 8 – *a* = 0, so *a* = 8.

For ** b = 6**, “11 divides

*1a96*” means that 1 –

*a*+ 9 – 6 = 4 –

*a*is divisible by 11, therefore 4 –

*a*= 0, so

*a*= 4.

**The numbers of type ***1a9b ***divisible by 44 are 1892 and 1496.**

**“4 divides 1x23y”** means that

*y*= 2 or

*y*= 6.

For ** y = 2**, “11 divides

*1×232*” means that (1 –

*x*+ 2 – 3 + 2) = (2 –

*x)*is divisible by 11 therefore 2 –

*x*= 0, so

*x*= 2.

For ** y = 6**, “11 divides

*1×236*” means that (1 –

*x*+ 2 – 3 + 6) = (6 –

*x)*is divisible by 11, therefore 6 –

*x*= 0, so

*x*= 6.

**The numbers of type ***1x23y ***divisible by 44 are 12232 and 16236.**

The fractions we were looking for are

^{1892}/_{12232} ,

^{1892}/

_{12232}

^{1496}/_{12232} ,

^{1496}/

_{12232}

^{1892}/_{16236} ,

^{1892}/

_{16236}

^{1496}/_{16236} .

^{1496}/

_{16236}

Done!

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