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# Nice Maths Question...

If lucky no. is defined as the no. whose sum of digits is 7, then lucky nos. are in sequence: 7,16,25,34... If 7=A1, 16=A2, 25=A3 and so on; find A65 and A325. (After adding digits once, you cannot add them more times for example:583 is not a lucky no. as you have to add its digits twice to get 7)

Note by Partho Paul
4 years, 6 months ago

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Stars and bars is applicable here.

For single digit: 7

(1 of them)

For two digits: we have $$\overline{a_1 a_2}$$, where $$a_1 + a_2=7$$, $$0 < a_1 \leq 7$$, $$0 \leq a_2 < 7$$ which gives $$16,25,34,43,52,61,70$$.

(7 of them)

For three digits: we have $$\overline{a_1 a_2 a_3}$$, where $$a_1 + a_2 + a_3 =7$$, $$0 < a_1 \leq 7$$, $$0 \leq a_2, a_3 < 7$$.

• When $$a_1 = 1$$, there's $${ 6+1 \choose 1 }$$ solution

• When $$a_2 = 2$$, there's $${ 5+1 \choose 1 }$$ solution

• ...

• when $$a_1 = 7$$, there's $${ 0+1 \choose 1 }$$ solution.

(28 of them)

For four digits: we have $$\overline{a_1 a_2 a_3 a_4}$$, where $$a_1 + a_2 + a_3 +a _4=7$$, $$0 < a_1 \leq 7$$, $$0 \leq a_2, a_3, a_4 < 7$$.

• When $$a_1 = 1$$, there's $${ 6+2 \choose 2 }$$ solution

• When $$a_1 = 2$$, there's $${ 5+2 \choose 2 }$$ solution

• ...

• When $$a_1 = 7$$, there's $${ 0+2 \choose 2 }$$ solution

(84 of them)

For five digits: we have $$\overline{a_1 a_2 a_3 a_4 a_5}$$, where $$a_1 + a_2 + a_3 +a _4 +a_5 =7$$, $$0 < a_1 \leq 7$$, $$0 \leq a_2, a_3, a_4, a_5 < 7$$.

• When $$a_1 = 1$$, there's $${ 6+3 \choose 3 }$$ solution

• When $$a_1 = 2$$, there's $${ 5+3 \choose 3 }$$ solution

• ...

• When $$a_1 = 7$$, there's $${ 0+3 \choose 3 }$$ solution

(210 of them)

Since $$1+7 + 28 < 65 < 1+7 + 28 + 84$$,

$$A_{65}$$ must be a four digit number. $$65-1-7-28 = 29$$.

$$29 - { 6+2 \choose 2 } = 1$$. So $$A_{65}$$ must the smallest four digit number with $$a_1 = 2$$.

$$\Rightarrow \boxed{ A_{65} = 2005 }$$

Similarly $$1+7 + 28 + 84 < 325 < 1+7 + 28 + 84 + 210$$,

$$A_{325}$$ must be a five digit number. $$325-1-7-28-84 = 205$$. Or $$325-1-7-28-84 -210 = -5$$

$$\Rightarrow A_{330} = 70000$$

$$\Rightarrow A_{329} = 61000$$

$$\Rightarrow A_{328} = 60100$$

$$\Rightarrow A_{327} = 60010$$

$$\Rightarrow A_{326} = 60001$$

$$\Rightarrow \boxed{ A_{325} = 52000 }$$

- 4 years, 6 months ago