# Divisibility by 4

Divisibility tests are often included in math competitions like the 2^{nd} MATH-Inic Vedic Mathematics National Challenge which will held online on April 24, 2022.

Most grade 5 learners know that in determining the divisibility of a number by 4, only the last two digits are considered. However, many students, and even adults, cannot explain why this is possible.

Also, when the last two digits are more than 40 (or beyond the regular 4 times table), some students find dividing directly by 4 difficult.

The Vedic Math provides two applicable sutras or word formulas:

1) **By Addition** and **By Subtraction **shows us why we can use the last two digits only and how we can avoid direct division. This sutra also enables us to develop divisibility rules for **any** number.

2) **The Ultimate and Twice the Penultimate **is seldom used but is applicable to divisibility by 4.

Adding or subtracting a multiple of a divisor to a number does not affect the divisibility of the number by that divisor. Since 100 is a multiple of 4, it follows that all multiples of 100 are also multiples of 4.

Now take a long number like 9876542 for example. **By Subtraction,** we can deduct from it 9876500, which is 98765 x 100 and leave only the last two digits 42. Now we only need to determine if 42 is divisible by 4.

For those who are not comfortable yet in dividing big numbers by 4, **By Addition and By Subtraction** also provides us with an easy way to determine the divisibility by 4 of any 2-digit number. Of, course we know that numbers ending in odd digits are not divisible by 4.

We only must remember the multiples of 4 below 20: 4, 8, 12 and 16 and the multiples of 20 up to 100: 20, 40, 60, 80 and 100. The general strategy is to deduct the next lower multiple of 20 from the number to be tested **or **add or subtract a multiple of 4 to get a multiple of 10.

Now let us consider the last two digits of the choices in our example:

- 52 – we can deduct 40 to get 12, add 8 to make it 60 or subtract 12 to get 40: divisible
- 64 – we can subtract 60 to get 4, add 16 to get 80 or deduct 4 to get 60: divisible
- 70 – by observation 70 is not a multiple of 20
- 48 – deduct 40 and get 8, add 12 to get 60 or subtract 8 to get 40: divisible
- 96 – adding 4 to make 100 is the easiest: divisible.

Another method is given by the sutra, **The Ultimate and Twice the Penultimate. **This means adding twice the second to the last digit (penultimate) to the last digit (ultimate). If the sum is a multiple of 4 then the number is divisible by 4.

- 52 – 2 + 2 x 5 = 12: divisible
- 64 – 4 + 2 x 6 = 16: divisible
- 70 – 0 + 2 x 7 = 14: not divisible
- 48 – 8 + 2 x 4 = 16: divisible
- 96 – 6 + 2 x 9 = 24: divisible

For those interested in participating in the 2^{nd} MATH-Inic Vedic Mathematics National Challenge, you can register thru an Area Facilitators in your school or division or to any of our Facilitators-At-Large:

Facilitators-At-Large (for Areas where there are no Facilitators)

Diana Lynn Luna

0917 535 4762

Janraey Carandang

0945 347 0857

Judy Mae Eseque

0906 072 0038

Gemma Dimaano

0933 859 1973

Exercises: Determine if the following numbers are divisible or not divisible by 4.

- 28
- 34
- 38
- 44
- 52
- 58
- 68
- 76
- 86
- 94

Answers to last week’s exercises

- 12 x 4 = 48
- 21 x 4 = 84
- 142 x 4 = 568
- 317 x 4 = 1268
- 1123 x 4 = 4492
- 3245 x 4 = 12980
- 21314 x 4 = 85256
- 23145 x 4 = 92580
- 342356 x 4 = 1369424
- 476324 x 4 = 1905296