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To find out, if a number is divisible by 7, take the last digit, double it, and subtract it from the rest of the number. If you get an answer divisible by 7 (including zero), then the original number is divisible by 7. If you don't know the new number's divisibility, you can apply the rule again.

If you want Proof of Rule for Divisibility By 7 , you may see at the bottom of the page.

Example 1 of Rule for Divisibility By 7

Find whether 343 is divisible by 7 or not.

Solution : Twice the last digit = 2 x 3 = 6; Rest of the number = 34 Subtracting, 34 - 6 = 28 is divisible by 7. ( 28 = 4 x 7) ∴ 343 is divisible by 7. Ans.

Example 2 of Rule for Divisibility By 7

Find whether 8965 is divisible by 7 or not.

Solution : Twice the last digit = 2 x 5 = 10; Rest of the number = 896 Subtracting, 896 - 10 = 886 To check whether 886 is divisible by 7 : Twice the last digit = 2 x 6 = 12; Rest of the number = 88 Subtracting, 88 - 12 = 66 is not divisible by 7. ( 66 = 9 x 7 + 3) ∴ 8965 is not divisible by 7. Ans.

Example 3 of Rule for Divisibility By 7

Find whether 49875 is divisible by 7 or not.

Solution : To check whether 49875 is divisible by 7 : Twice the last digit = 2 x 5 = 10; Rest of the number = 4987 Subtracting, 4987 - 10 = 4977

To check whether 4977 is divisible by 7 : Twice the last digit = 2 x 7 = 14; Rest of the number = 497 Subtracting, 497 - 14 = 483

To check whether 483 is divisible by 7 : Twice the last digit = 2 x 3 = 6; Rest of the number = 48 Subtracting, 48 - 6 = 42 is divisible by 7. ( 42 = 6 x 7 )

∴ 49875 is divisible by 7. Ans.

Example 4 of Rule for Divisibility By 7

Find whether 987651 is divisible by 7 or not.

Solution : To check whether 987651 is divisible by 7 : Twice the last digit = 2 x 1 = 2; Rest of the number = 98765 Subtracting, 98765 - 2 = 98763

To check whether 98763 is divisible by 7 : Twice the last digit = 2 x 3 = 6; Rest of the number = 9876 Subtracting, 9876 - 6 = 9870

To check whether 987 is divisible by 7 : Twice the last digit = 2 x 7 = 14; Rest of the number = 98 Subtracting, 98 - 14 = 84

To check whether 84 is divisible by 7 : Twice the last digit = 2 x 4 = 8; Rest of the number = 8 Subtracting, 8 - 8 = 0 is divisible by 7.

∴ 987651 is divisible by 7. Ans.

Example 5 of Rule for Divisibility By 7

Find whether 986953 is divisible by 7 or not.

Solution : To check whether 986953 is divisible by 7 : Twice the last digit = 2 x 3 = 6; Rest of the number = 98695 Subtracting, 98695 - 6= 98689

To check Divisibility By 7 of 98689 : Twice the last digit = 2 x 9 = 18; Rest of the number = 9868 Subtracting, 9868 - 18 = 9850

To check Divisibility By 7 of 985 :Twice the last digit = 2 x5 = 10; Rest of the number = 98 Subtracting, 98 - 10 = 88

To check Divisibility By 7 of 88 :Twice the last digit = 2 x 8 = 16; Rest of the number = 8 Subtracting, 8 - 16 = -8 is not divisible by 7.

∴ 986953 is not divisible by 7. Ans.

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Let D_{1} be the units' digit and D_{2} be the rest of the number of D. i.e. D = D_{1} + 10D_{2}

We have to prove (i) if D_{2} - 2D_{1} is divisible by 7, then D is also divisible by 7

and (ii) if D is divisible by 7, then D_{2} - 2D_{1} is also divisible by 7.

Proof of (i) : D_{2} - 2D_{1} is divisible by 7 ⇒ D_{2} - 2D_{1} = 7k where k is any natural number. Multiplying both sides by 10, we get 10D_{2} - 20D_{1} = 70k Adding D_{1} to both sides, we get (10D_{2} + D_{1}) - 20D_{1} = 70k + D_{1} ⇒ (10D_{2} + D_{1}) = 70k + D_{1} + 20D_{1} ⇒ D = 70k + 21D_{1} = 7(10k + 3D_{1}) = a multiple of 7. ⇒ D is divisible by 7. (proved.)

Proof of (ii) : D is divisible by 7 ⇒ D_{1} + 10D_{2} is divisible by 7⇒ D_{1} + 10D_{2} = 7k where k is any natural number. Subtracting 21D_{1} from both sides, we get 10D_{2} - 20D_{1} = 7k - 21D_{1} ⇒ 10(D_{2} - 2D_{1}) = 7(k - 3D_{1}) ⇒ 10(D_{2} - 2D_{1}) is divisible by 7 Since 10 is not divisible by 7, (D_{2} - 2D_{1}) is divisible by 7. (proved.)

In a similar fashion, we can prove the divisibility rule for any prime divisor.

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