Question

How many numbers between 100 and 500 are exactly divisible by 4.

Answer 1

Answer:

the question “••• between 100 and 500 are divisible by 4,7 and 9 simultaneously? “

Solution (According to the question as meant )

Let the number be x.

x=0 (mod 4)

x=0 (mod 7)

x =0 (mod 9)

This is a Chinese Remainder Theorem.

x=LCM of (4,7,9)=252.

In Parametrical form

x=0 + 252 t where t is any integer.

Let the t=2 then,

x=252*2=504 Since >500 , so discarded.

Therefore is one NUMBER (252)

SECOND SOLUTION

Number starts ends Number of multiples

*** 4*** *** 104 **496***(496–104)/4+1=99

***7*** ***105 **497***(497–105)/7+1=57

***9*** ***108 **495***(495–108)/9+1=44

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Total 200

DEDUCT Common Multiples

4×7 (28)****112 **476******************=14

4×9 (36)****108 **468 *****************=11

9×7 (63)****126 **441 *******************=6

Total 31×2=62

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Total Multiples

(99+57+44)- 62=138 Answer.