Divisible by 44

How many fractions of type  1a9b/1x23y can be simplified by 44?

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 ,

1496/12232 ,

1892/16236 ,

1496/16236 .

Done!

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