16

Divisibility by 16

    Method 1:

  • If the number formed by the last four digits at the thousands, hundreds ,tens and unit place is divisible by 16 then the number is divisible by 16.

  • Method 2:
    1. The digit at the thousands place is multiplied by 8
    2. The digit at the hundreds place is multiplied by 4.
    3. The digit at the tens place is multiplied by 10.
    4. The result of step 1, 2 and 3 is added to the digit at the unit place.
    5. Repeat the process if you get a large number.
    6. The result is divided by 16. If the remainder is zero then the number is divisible else the remainder is the remainder of the original number.

Proof:

Let the number be abcde.

abcde can be written as
a×10000 + b×1000 + c×100 + d×10 + e

which can also be written as
a×16×625 + b×(16×62+8) + c×(16×6+4) + d×10 + e

Every power of 10 i.e. 10x is divisible by 16 if x>3.

If the number formed by the last four digits is divisible by 16 then the number is divisible by 16.

So, a×16×625 + b×(16×62+8) + c×(16×6+4) + d×10 + e

can be written as
a×16×625 + b×(16×62) + 8b + c×(16×6)+ 4c + d×10 + e
Products containing factors of 16 are divisible.
The whole sum is divisible if b×8 + c×4 + d×10 + e is divisible by 16.

The number is divisible by 16 if 8b + 4c + 10d + e is divisible by 16.
Remainder

  1. The number formed by the last four digits when divided by 16 gives a remainder and this is the remainder of the original number.

  2. If the number formed by the last four digits is abcd. Then the remainder is the remainder of the number 8a+4b+10c+d when divided by 16.

Examples

  1. Is 348 divisible by 16.
    (3×4+4×10+8)=60. The remainder when 60 is divided by 16 is 12. Hence the remainder of the original number is 12.

  2. Is 1344 divisible by 16.
    The number formed by the last four digits is 1344 and 1×8+3×4+4×10+4=64 and it is divisible by 16. Hence, the number is divisible by 16.


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