In this note, we want to look at how a general divisibility test for \(N\) could be obtained. For each \(N\), we want to find a corresponding \(n\) such that \( N \mid 10x + y \Leftrightarrow N \mid x + n y \).
This is a generalization of the famous divisibility test of 7, in which we multiply the last digit by 2 and then add it to the rest of the number. Namely, 10x+y is a multiple of 7 if and only if x+2y is a multiple of 7. This gives us a quick way to find if the number 123456 is a multiple of 7, by checking 12345+12=12357, and then checking 1235+14=1249 and then checking 124+18=142 and then checking 14+4=18, which we know is not a multiple of 7.
Let number be 10x+y, e.g. 93468 as 10∗9346+8, so x=9346 and y=8.
Nn.
7:~~~ 10x + y~~~ \implies~~~x+5y~~~.... n=5.\\~~~~~~~~ OR~~~ 10x + y \impliesx- 2y ~~~....n=-2\\~~~~~~~~ OR ~~~~~~~~ \text{
10x + y has the same remainder when divided by 7 as 3x + y.} \\
~~~~~~~~Say~~ 371:~~~ 3×3 + 7 = 16~
remainder~ 2,~ and~ 2×3 + 1 = 7.~\therefore 7|371.
13: 10x+y ⟹ x+4y n=417: 10x+y ⟹ x+5y n=519: 10x+y ⟹ x+2y n=223: 10x+y ⟹ x+7y n=729: 10x+y ⟹ x+3y n=331: 10x+y ⟹ x−3y n=−337: 10x+y ⟹ x+11y n=1141: 10x+y ⟹ x−4y n=−443: 10x+y ⟹ x+13y n=1347: 10x+y ⟹ x−14y n=−1453: 10x+y ⟹ x+16y n=1659: 10x+y ⟹ x+6y n=6
To illustrate , Test 41|2829:-x=282, y=9 and for 41, n=4.∴ 282-4 * 9=246 . Repeat for 246. 24-4 * 6=0 so 41|246.Repeat several times for a big number.Say 41∣28648914 x=2864891, y=4, ∴ 2864891−4∗4=2864875,x=286487,y=5, ∴ 286487−4∗5=286467,x=28646,y=7, ∴ 28646−4∗7=28618,x=2861, y=8, ∴ 2861−4∗8=2829,x=282, y=9,, ∴ 282−4∗9=246,x=24, y=6,, ∴ 24−4∗6=0. Note, final result will be 0 if the number is divisible.∴ 41∣28648914. I am adding Calvin Lin’s comment below. It is nice general result.For given value of N that satisfies gcd(N,10) = 1,corresponding value of n is just 10−1(modN).10x+y≡0(modN)⇔x+10−1y≡0(modN).
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Do you know why this is true? IE 7∣10x+y⇔7∣x+5y?
Also, for a given value of N, what would the corresponding value of n be? Why?
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Yes. From theory of numbers, we can see why it works. But these are simple short cut notes. Is it of much use now with our calculator?? But for solving related problem, we may just use it in stead of going into full proof. Just like saying rationalize the denominator and put the result with out intermediate steps.
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There's a simple one line proof of that.
7∣10x+y⟺10x+y≡0(mod7)÷5⟺×550x+5y≡0(mod7)⟺x+5y≡0(mod7)⟺7∣x+5y
The ×5 part is for the forward direction and the ÷5 part is for the backward direction in the proof. Note that division by 5 is well defined here since gcd(5,7)=1.
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N which satisfies gcd(N,10)=1, the corresponding value of n is just 10−1(modN).
Great! So, for a given value of10x+y≡0(modN)⇔x+10−1y≡0(modN)
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Thank so much. It is a lot of improvement .