Given-N-in-base-10-is-158b687a-where-b-and-a-are-digits-a-lt-b-Given-N-2-a-4-1-Determinate-couples-a-b-such-that-11-divise-N-2-Determinate-couples-a-b-such-that-3-and-25-divise-N-thanks

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Question Number 121568 by mathocean1 last updated on 09/Nov/20

$$\mathrm{Given}:\mathrm{N}\mathrm{in}\mathrm{base}10\mathrm{is}$$$$158\mathrm{b}687\mathrm{a}\mathrm{where}\mathrm{b}\mathrm{and}\mathrm{a}\mathrm{are}$$$$\mathrm{digits}(\mathrm{a}<\mathrm{b}).\mathrm{Given}:$$$$\mathrm{N}\equiv 2+\mathrm{a}\left[4\right].$$$$1)\mathrm{Determinate}\mathrm{couples}$$$$(\mathrm{a};\mathrm{b})\mathrm{such}\mathrm{that}11\mathrm{divise}\mathrm{N}.$$$$2)\mathrm{Determinate}\mathrm{couples}(\mathrm{a};\mathrm{b})$$$$\mathrm{such}\mathrm{that}3\mathrm{and}25\mathrm{divise}\mathrm{N}.$$$$\mathrm{thanks}.$$

Answered by kolos last updated on 17/Nov/20

$$$$$$1)jf11diviseN,thenN\equiv 0mod\left(11\right)$$$$15806870+10000b+a\equiv 0mod\left(11\right)$$$$2+b+a\equiv omod\left(11\right)$$$$a+b\equiv 9mod11$$$$(a,b)=(0,9)(1,8)(2,7)\dots (8,1)(9,0)$$$$$$$$2)ifNismultipleof25,therefore$$$$thelasttwodigitsare00,25,50or75$$$$$$$$asthelasttwodigitsofNare7a,then$$$$a=5$$$$$$$$tobemultipleof3,thesumofthe$$$$digitsareamultipletoo$$$$$$$$then$$$$1+5+8+b+6+8+7+5\equiv 0mod3$$$$b+40\equiv 0mod3$$$$b\equiv 2mod3$$$$b=2,5,8$$$$$$$$(a,b)=(5,2)(5,5)(5,8)$$$$$$$$Ihope{i}^{\prime }vehelped:)$$$$$$

Commented by mathocean1 last updated on 19/Nov/20

$$thankyouverymuchsir!$$

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