DIVISION TRICKS

DIVISION TRICKS        

DIVISIBLE BY 2:

WHEN  THE LAST DIGIT OF ANY NUMBER IS EVEN (2,4,6,8,0)

SINCE THE LAST DIGIT OF 23578, WHICH IS 8, IS DIVISIBLE BY 2, THEN 23578 IS

DIVISIBLE   BY  2.

THE LAST DIGIT OF 2365 IS 5. SINCE 5 IS NOT DIVISIBLE BY 2, 2365 IS NOT

DIVISIBLE  BY  2.

DIVISIBLE BY 3:

(WHEN THE SUM OF ALL NUMBERS IS DIVISIBLE BY 3)

LET US SEE IF 231 IS DIVISIBLE BY 3. SUMMING UP THE DIGITS OF 231 WE GET

2 + 3 + 1 = 6. SINCE 6 IS DIVISIBLE BY 3 THEN 231 IS DIVISIBLE BY 3.

LET US CHECK IF 23574 IS DIVISIBLE BY 3. SUMMING UP THE DIGITS OF 23574

GIVES US  2 + 3 + 5 + 7 + 4 = 21.  SINCE 21 IS DIVISIBLE BY 3, THEN 23574

IS DIVISIBLE BY 3. BUT SUPPOSE THAT YOU COULD NOT REMEMBER THAT 21IS DIVISIBLE BY 3.

THEN YOU COULD SUM UP THE DIGITS OF 21 TO GET 2+1 = 3.

CLEARLY 3 IS DIVISIBLE BY 3. SO 21 IS DIVISIBLE BY 3 AND THEREFORE  23574

IS DIVISIBLE BY 3.

IS 27934576592477693217 DIVISIBLE BY 3? LET US SUM UP THE DIGITS AGAIN

2+7+9+3+4+5+7+6+5+9+2+4+7+7+6+9+3+2+1+7 = 105.

IF WE ARE NOT SURE IF 105 IS DIVISIBLE BY THREE WE CAN SUM UP ITS DIGITS.

SO 1 + 0 + 5 = 6. NOW WE KNOW 6 IS DIVISIBLE BY 3, SO 105 IS DIVISIBLE

BY 3 AND THEREFORE 27934576592477693217 IS DIVISIBLE BY 3.

THE NUMBER 16 IS NOT DIVISIBLE BY 3 SINCE 1 + 6 = 7 IS NOT DIVISIBLE BY

3.

LET US TRY 23761. IT'S DIGITS SUM TO 2 + 3 + 7 + 6 + 1 = 19 AND 19 IS

NOT DIVISIBLE BY 3 SINCE 1 + 9 = 10 IS NOT DIVISIBLE BY 3.

DIVISIBLE BY 4:

(WHEN LAST 2 DIGITS ARE 00 OR DIVISBLE BY 4)

THE NUMBER 116 IS DIVISIBLE BY 4 SINCE THE LAST TWO DIGITS OF 116 IS 16AND

16 IS DIVISIBLE BY 4.

HOW ABOUT THE NUMBER 1352? THE LAST TWO DIGITS OF 1352 IS 52.

 NOW 4 WILL DIVIDE 52 (52 _ 4 = 13). SO 4 DIVIDES 1352.

WILL 4 DIVIDE 125487100?

YES SINCE THE LAST TWO DIGITS ARE 00.

DIVISIBLE BY 5

(WHEN NUMBR END WITH 5 OR 0)

IT IS EASY TO SEE THAT 5 WILL DIVIDE 25, 40, 14520, 785425, AND789875412545,

SINCE EACH ONE ENDS IN  0 OR  5.

DIVISIBLE BY 6:

(WHEN THE NUMBER IS DIVISIBLE BY BOTH 2 AND 3)

TO CHECK IF 132 IS DIVISIBLE BY 6 REQUIRES TWO STEPS. FIRST IS 132DIVISIBLE BY 2?

YES SINCE THE LAST DIGIT IS DIVISIBLE BY 2.

NEXT WE SUM UP ITS DIGITS:1 + 3 + 2 = 6. SINCE 6 IS DIVISIBLE BY 3,

132 IS DIVISIBLE BY 3. THEREFORE 132 IS DIVISIBLE BY 6.

LET US TRY 1764. CLEARLY 1764 IS DIVISIBLE BY 2 AND IT IS DIVISIBLE BY 3

SINCE 1 + 7 + 6 + 4 = 18 = 9. SO 1764 IS DIVISIBLE BY 6.

THE NUMBER 1371 IS NOT DIVISIBLE BY 6 SINCE 1371 IS NOT DIVISIBLE BY 2.

NOTE THAT IT IS DIVISIBLE BY 3, 1+3+7+1 = 12, BUT THAT DOES NOT MATTER

SINCE IT IS NOT DIVISIBLE BY 2.

THE NUMBER 1678 IS NOT DIVISIBLE BY 6 SINCE IT IS NOT DIVISIBLE BY 3. THIS

CAN BE SEEN SINCE 1+6+7+8 = 22. IT DOES NOT MATTER THAT IT IS DIVISIBLE

BY 2. IT HAS TO BE DIVISIBLE BY BOTH.

WE CAN SEE THAT 3211 IS NOT DIVISIBLE BY 6 SINCE IT IS NOT DIVISIBLE BY

EITHER 2 OR 3.

DIVISIBLE BY 7:

LET US START WITH SOME EASY EXAMPLES.

FIRST 14 IS DIVISIBLE BY 7 SINCE

1 - 2(4) = 1 - 8 = -7 AND -7 IS DIVISIBLE BY 7.

NEXT 21 IS DIVISIBLE BY

7 SINCE 2 - 2(1) = 2 - 2 = 0 AND 0 IS DIVISIBLE BY 7.

IS 924 DIVISIBLE BY 7?

WE APPLY THE RULE TO GET 92-2(4) = 92-8 = 84.

IF WE ARE NOT SURE WHETHER 7 DIVIDES 84 WE CAN APPLY THE RULE AGAIN.

2

DOING THIS GIVES US 8-2(4) = 8-8 = 0 AND 0 IS DIVISIBLE BY 7. SO 924

IS DIVISIBLE BY 7.

OUR NEXT EXAMPLE IS TO DETERMINE IF 2632 IS DIVISIBLE BY 7.

 APPLY THE RULE WE GET 263 - 2(2) = 263 - 4 = 259.

 APPLYING THE RULE AGAIN

25 - 2(9) = 25 - 18 = 7. SO 7 DIVIDES 2632.

IS 7532 DIVISIBLE BY 7? LET US APPLY THE RULE: 753-2(2) = 753-4 = 749.

APPLYING THE RULE AGAIN WE GET 74 - 2(9) = 74 - 18 = 56. IF WE DO NOT

RECOGNIZE THAT 56 = 7 _ 8 WE CAN APPLY THE RULE ANOTHER TIME TO GET

5 - 2(6) = 5 - 12 = -7. SO 7 DIVIDES 7532.

DOES 7 DIVIDE 72? WELL 7 - 2(2) = 7 - 4 = 3, SO 7 DOES NOT DIVIDE 72.

DIVISIBLE BY  8:

(WHEN LAST 3 DIGITS ARE 000 OR DIVISBLE BY 8)

THIS IS A LITTLE TRICKER TO APPLY SINCE YOU HAVE TO LOOK AT THE LAST THREE DIGITS.

FOR EXAMPLE, DOES 8 DIVIDE 3120? SINCE 8 DIVIDES 120, 8 DIVIDES

3120. BUT YOU HAVE TO KNOW THAT 8 DIVIDES 120 TO APPLY THIS TRICK. SO

NORMALLY I JUST DIVIDE THE NUMBER BY 2 AND IF I CAN DO THIS THREE TIMES

THE NUMBER IS DIVISIBLE BY 8. OF COURSE I WOULD ALREADY KNOW THAT SINCE

I HAVE THE QUOTIENT. THERE ARE SITUATIONS WHERE THIS RULE IS ACTUALLY

USEFUL FOR ME.

FIRST IF THE NUMBER ENDS IN THREE 0'S I KNOW IT IS DIVISIBLE BY 8. FOR

EXAMPLE, THE NUMBER 1548000 IS DIVISIBLE BY 8. THE OTHER TIME I _ND

THIS TRICK USEFUL IS IF THE THIRD DIGIT IS A 0.

THEN I CAN JUST CHECK TO SEE

IF THE LAST TWO DIGITS ARE DIVISIBLE BY 8. AS AN EXAMPLE LET US LOOK AT

547024. TO CHECK THIS NUMBER FOR DIVISIBILITY BY 8 WE LOOK AT 024 OR AS

WE USUALLY WRITE IT 24. SINCE 24 IS DIVISIBLE BY 8, SO IS 547024.

DIVISIBLE BY 9:

IS 234 DIVISIBLE BY 9? YES SINCE 2+3+4 = 9 AND 9 IS DIVISIBLE BY 9.

ALSO

31275 IS DIVISIBLE BY 9 SINCE 3 + 1 + 2 + 7 + 5 = 18 AND 18 IS DIVISIBLE

BY 9 (1 + 8 = 9).

LET US TRY 573. SINCE 5 + 7 + 3 = 15 AND 15 IS NOT DIVISIBLE BY 9, 573

IS NOT DIVISIBLE BY 9.

DIVISIBLE BY 10:

THE FOLLOWING ARE ALL DIVISIBLE BY 10 SINCE THEY END IN A 0: 20, 25640,

87951880,AND 603020105040708090201020.

DIVISIBLE BY 11:

(SUM OF ALL NUMBER ON ODD POSITION - SUM OF ALL NUMBER ON EVEN POSITION)

DOES 11 DIVIDE 143?

TO CHECK WE DO THE FOLLOWING CALCULATION: (1+3)-

4 = 4 - 4 = 0. SINCE 11 DIVIDES 0 IT DIVIDES 143.

TO CHECK TO SEE IF 11 DIVIDES 2607 WE MAKE THE CALCULATION: (2 + 0) -

(6 + 7) = 2 - 13 = -11. SINCE 11 DIVIDES -11 IT DIVIDES 2607.

NOW TRY 38236. SO (3 + 2 + 6) - (8 + 3) = 11 - 11 = 0. SO 11 DIVIDES

38236.

FINALLY DOES 11 DIVIDE 3042149? WELL SINCE 11 DIVIDES (3+4+1+9)-

(0 + 2 + 4) = 17 - 6 = 11 IT DIVIDES 3042149.

DIVISIBLE BY12:

(IT HAS TO BE DIVISIBLE BY BOTH 3 AND 4)

IS 132 DIVISIBLE BY 12? IF IT IS IT HAS TO BE DIVISIBLE BY BOTH 3 AND 4. SINCE

1 + 3 + 2 = 6 IT IS DIVISIBLE BY 3 AND SINCE 32 IS DIVISIBLE BY 4, 132 IS

DIVISIBLE BY 12.

WHAT ABOUT 264? AGAIN WE TEST FOR DIVISIBILITY BY 3 AND 4. SINCE 2 +

6 + 4 = 12 IS DIVISIBLE BY 3 AND 64 IS DIVISIBLE BY 4, 264 IS DIVISIBLE BY

12.

CAN WE DIVIDE 1800 BY 12? FIRST WE ADD THE DIGITS OF 1800 TO GET

1+8+0+0 = 9. SO 1800 IS DIVISIBLE BY 3. THE LAST TWO DIGITS OF 1800

ARE 00, SO 1800 IS DIVISIBLE BY 4. THEREFORE 1800 IS DIVISIBLE BY 12.

DIVISIBLE BY 13:

IS 65 DIVISIBLE BY 13? BY THE RULE WE COMPUTE

6 - 9(5) = 6 - 45 = -39:

SINCE 39 IS DIVISIBLE BY 13, 65 IS DIVISIBLE BY 13.

LETS TRY 377.

37 - 9(7) = 37 - 63 = -26

SO 377 IS DIVISIBLE BY 13, SINCE 26 IS DIVISIBLE BY 13.

IS 2041 DIVISIBLE BY 13?

204 - 9(1) = 195

19 - 9(5) = 19 - 45 = -26

SO 2041 IS DIVISIBLE BY 13.


Source: http://helperhelp.blogspot.in/