Two-different-prime-numbers-between-4-and-18-are-chosen-When-their-sum-is-subtracted-from-their-product-then-a-number-x-is-obtained-which-is-a-multiple-of-17-Find-the-sum-of-digits-of-number-x-

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Question Number 19631 by Tinkutara last updated on 13/Aug/17

$$\mathrm{Two}\mathrm{different}\mathrm{prime}\mathrm{numbers}\mathrm{between}$$$$4\mathrm{and}18\mathrm{are}\mathrm{chosen}.\mathrm{When}\mathrm{their}\mathrm{sum}\mathrm{is}$$$$\mathrm{subtracted}\mathrm{from}\mathrm{their}\mathrm{product}\mathrm{then}\mathrm{a}$$$$\mathrm{number}x\mathrm{is}\mathrm{obtained}\mathrm{which}\mathrm{is}\mathrm{a}$$$$\mathrm{multiple}\mathrm{of}17.\mathrm{Find}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{digits}\mathrm{of}$$$$\mathrm{number}x.$$

Answered by ajfour last updated on 13/Aug/17

$$\mathrm{let}\mathrm{p}\mathrm{and}\mathrm{q}.$$$$\mathrm{pq}-(\mathrm{p}+\mathrm{q})=17\mathrm{n}$$$$\mathrm{p},\mathrm{q}\mathrm{can}\mathrm{be}\mathrm{among}5,7,11,13,17$$$$\mathrm{if}\mathrm{p}=17,\mathrm{q}=13$$$${15}^2-4-30=191\ne 17\mathrm{n}$$$$\mathrm{if}\mathrm{p}=17,\mathrm{q}=11$$$$196-9-28=159\ne 17\mathrm{n}$$$$\mathrm{if}\mathrm{p}=17,\mathrm{q}=7$$$$144-25-24=95\ne 17\mathrm{n}$$$$\mathrm{if}\mathrm{p}=17,\mathrm{q}=5$$$$85-22=63\ne 17\mathrm{n}$$$$\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots .$$$$\mathrm{if}\mathrm{p}=13,\mathrm{q}=11$$$$143-24=119=17\times 7$$$$\mathrm{if}\mathrm{p}=13,\mathrm{q}=7$$$$91-20=71\ne 17\mathrm{n}$$$$\mathrm{if}\mathrm{p}=13,\mathrm{q}=5$$$$65-18=47\ne 17\mathrm{n}$$$$\mathrm{if}\mathrm{p}=11,\mathrm{q}=7$$$$77-18=59\ne 17\mathrm{n}$$$$\mathrm{if}\mathrm{p}=11,\mathrm{q}=5$$$$55-16=39\ne 17\mathrm{n}$$$$\mathrm{if}\mathrm{p}=7,\mathrm{q}=5$$$$35-12=23\ne 17\mathrm{n}$$$$\mathrm{Hence}\mathrm{x}=17\times 7=119$$$$\mathrm{Sum}\mathrm{of}\mathrm{its}\mathrm{digits}=11.$$

Commented by Tinkutara last updated on 13/Aug/17

$$\mathrm{So}\mathrm{totally}\mathrm{hit}\mathrm{and}\mathrm{trial}\mathrm{method}\mathrm{is}$$$$\mathrm{applicable}?$$

Commented by ajfour last updated on 13/Aug/17

$$\mathrm{dint}\mathrm{know}\mathrm{a}\mathrm{decent}\mathrm{way}\mathrm{out}..$$

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