Olympiad Maths Prep

Track / Stage 5 / 59 of 400 #659 of 2000

Problem 659

AIME late
Number theory Difficulty 5.2 Find the answer

## Task 1 - 301211

Determine whether there exist natural numbers a,b,c,da, b, c, d that satisfy the following two conditions (1) and (2):

a⋅b⋅c⋅d=111111111111a+b+c+d<11111 \begin{aligned} a \cdot b \cdot c \cdot d & =111111111111 \\ a+b+c+d & <11111 \end{aligned}

If this is the case, provide such numbers.

Official solution

It suffices to provide an example of natural numbers a,b,c,da, b, c, d and to confirm (1), (2) for the given numbers. One such example is

a=37,b=3⋅11⋅13=429,c=7⋅101=707,d=9901 a=37, \quad b=3 \cdot 11 \cdot 13=429, \quad c=7 \cdot 101=707, \quad d=9901

which can be verified by calculating a+b+c+d=11074a+b+c+d=11074 and

3⋅37=111,7⋅11⋅13=1001,101⋅9901=1000001 3 \cdot 37=111, \quad 7 \cdot 11 \cdot 13=1001, \quad 101 \cdot 9901=1000001

thus

3⋅7⋅11⋅13⋅37=111⋅1001=111111 3 \cdot 7 \cdot 11 \cdot 13 \cdot 37=111 \cdot 1001=111111

a⋅b⋅c⋅d=3⋅7⋅11⋅13⋅37⋅101⋅9901=111111⋅1000001=111111111111 a \cdot b \cdot c \cdot d=3 \cdot 7 \cdot 11 \cdot 13 \cdot 37 \cdot 101 \cdot 9901=111111 \cdot 1000001=111111111111

Taken from [5][5]

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.